The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes

*Yasuhiko Kamiyama*

#### **Abstract**

As a mathematical model for cycloalkenes, we consider equilateral polygons whose interior angles are the same except for those of the both ends of the specified edge. We study the configuration space of such polygons. It is known that for some case, the space is homeomorphic to a sphere. The purpose of this chapter is threefold: First, using the *h*-cobordism theorem, we prove that the above homeomorphism is in fact a diffeomorphism. Second, we study the best possible condition for the space to be a sphere. At present, only a sphere appears as a topological type of the space. Then our third purpose is to show the case when a closed surface of positive genus appears as a topological type.

**Keywords:** cycloalkene, polygon, configuration space, *h*-cobordism theorem, closed surface

#### **1. Introduction**

The configuration space of mechanical linkages in the Euclidean space of dimension three, also known as polygon space, is the central objective in topological robotics. The linkage consists of *n* bars of length *l*1, ⋯, *ln* connected by revolving joints forming a closed spatial polygonal chain.

The polygon space is quite important in various engineering applications: In molecular biology they describe varieties of molecular shapes, in robotics they appear as spaces of all possible configurations of some mechanisms, and they play a central role in statistical shape theory.

Mathematically, these spaces are also very interesting: The symplectic structure on the polygon space was studied in the seminal paper [1]. The integral cohomology ring was determined in [2] applying methods of toric topology. We refer to [3] for an excellent exposition with emphasis on Morse theory.

Recently, mathematicians are interested in a mathematical model for monocyclic hydrocarbons. The model is defined by imposing conditions on the interior angles of a polygon. The configuration space of such polygons corresponds in chemistry to the conformations of all possible shapes of a monocyclic hydrocarbon. Hence the configuration space is interesting both in mathematics and chemistry.

In order to give more detailed account, recall that monocyclic hydrocarbons are classified into two types: One is saturated type, and the other is unsaturated type. Mathematicians constructed a mathematical model for each type. We summarize

the correspondence between chemical and mathematical terminologies in the following **Table 1**.

Below we explain **Table 1**.

	- The mathematical model for cycloalkanes is the equilateral and equiangular polygons. Let M*n*ð Þ*θ* be the configurations of such *n*-gons with interior angle *θ*. The study of the topological type of M*n*ð Þ*θ* originated in [4]. See the next item for more details.
	- The topological type of M4ð Þ*θ* and M5ð Þ*θ* was determined in [4] for arbitrary *θ*, and that of M6ð Þ*θ* was determined in [5] for arbitrary *θ*. The paper [4] also determined the topological type of M7ð Þ*θ* for the case that *θ* is the ideal tetrahedral bond angle, i.e. *θ* ¼ arccos � <sup>1</sup> 3 ≈109*:*47°. The result was generalized in [6] for generic *θ*.
	- Hereafter, for simplicity, we consider only the monocyclic unsaturated hydrocarbons that contain exactly one multiple bond.
	- It is not mathematically important whether the multiple bond is a double or triple bond. Hence we assume that the multiple bond is a double bond.


#### **Table 1.**

*The correspondence between chemical and mathematical terminologies.*

**Figure 1.** *Cyclohexane (6-membered cycloalkane).*

*The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*

**Figure 2.** *Cyclohexene (6-membered cycloalkene).*


On the other hand, as a combinatorial result, the necessary and sufficient condition for M*n*ð Þ*θ* and *Cn*ð Þ*θ* to be non-empty was proved in [9]. (See Theorem 3 about the result for *Cn*ð Þ*θ* .)

As stated in the last item of the above ii, we do not have enough information about the topology of *Cn*ð Þ*θ* . The purpose of this chapter is to obtain systematic information about *Cn*ð Þ*θ* . More precisely, we study the following:

**Problem 1.** (i) We prove that the above homeomorphism in [8] is in fact a diffeomorphism.

(ii) We study the best possible value about the above *θ*<sup>0</sup> in [8].

(iii) At present, only a sphere appears as a topological type of *Cn*ð Þ*θ* . We determine the topological type of *C*6ð Þ*θ* for all *θ*. The result shows that for some *θ*, *C*6ð Þ*θ* is a closed surface of positive genus.

This chapter is organized as follows. In §2, we state our main results. In §3-§5, we prove them. In §6, we state the conclusions.

#### **2. Main results**

We give the definition of the configuration space. Let *θ* be a real number satisfying 0 ≤*θ* ≤ *π*. We set

$$\mathbf{C}\_{n}(\theta) \coloneqq \left\{ P = (\boldsymbol{u}\_{1}, \dots, \boldsymbol{u}\_{n}) \in \left(\mathbb{S}^{2}\right)^{n} \; | \; \text{the following i, ii and iii hold} \right\}. \tag{1}$$
  $\text{i. } \boldsymbol{u}\_{1} = (\mathbf{1}, \mathbf{0}, \mathbf{0}) \text{ and } \boldsymbol{u}\_{n} = (-\cos\theta, -\sin\theta, \mathbf{0}).$ 

i.  $\sum\_{i=1}^{n} \mu\_i = 0$ .

iii. *ui* h i , *ui*þ<sup>1</sup> ¼ � cos *θ* for 1≤ *i*≤ *n* � 3, where , h i denotes the standard inner product on <sup>3</sup> .

About the conditions in (1), the following explanations are in order. (See **Table 1** for chemical terminologies.)


**Remark 2.** In some papers, *Cn*ð Þ*θ* is defined as

$$A\_n(\theta) / \text{SO}(\mathfrak{Z}),\tag{2}$$

where we set

*An*ð Þ*<sup>θ</sup>* <sup>≔</sup> ð Þ *<sup>u</sup>*1, <sup>⋯</sup>, *un* <sup>∈</sup> *<sup>S</sup>*<sup>2</sup> � �*<sup>n</sup>* <sup>j</sup> ð Þ<sup>1</sup> ii, iii and the conditionh i *un*, *<sup>u</sup>*<sup>1</sup> ¼ � cos *<sup>θ</sup>* hold � �*:*

Let *SO*ð Þ3 act on *An*ð Þ*θ* diagonally. Then for an element ð Þ *u*1, ⋯, *un* ∈ *An*ð Þ*θ* , we may normalize *u*<sup>1</sup> and *un* to be as in (1) i. Hence (2) in fact coincides with (1).

The following result is known.

**Theorem 3** ([9], Theorems A and B). *(i) For n*≥4*, we have Cn*ð Þ*θ* 6¼ ∅ *if and only if θ belongs to the following interval:*

$$\begin{cases} \left[ 2 \arcsin \frac{1}{n-1}, \frac{n-2}{n} \pi \right], & \text{if } n is odd, \\\left[ 0, \frac{n-2}{n} \pi \right], & \text{if } n \text{ is even.} \end{cases} \tag{3}$$

*(ii). Let a be an endpoint of the intervals in* (3). *Then we have Cn*ð Þ¼ *a* f g *one point :* **Example 4.** For *n* ¼ 4 or 5, the following results hold, where we omit the cases which can be read from Theorem 3.

i. For 0 <*θ* < *<sup>π</sup>* <sup>2</sup>, we have *C*4ð Þ¼ *θ* f g two points .

ii. The topological type of *C*5ð Þ*θ* is given by the following **Table 2**.

Here we define *η*<sup>1</sup> and *η*<sup>2</sup> to be the following **Figures 3** and **4**, respectively. The proof of the example will be given at the end of §5.

In [8], the following proposition is proved using the implicit function theorem. **Proposition 5** ([8], Proposition 1). *There exists θ*<sup>0</sup> *such that for all θ* ∈ *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � �*, the system of equations defined by* (1) i*,* ii *and* iii *intersect transversely. Hence for such θ, Cn*ð Þ*θ carries a natural differential structure.*


**Table 2.** *The topological type of C*5ð Þ*θ .* *The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*

**Figure 3.** *The space η*1*.*

The main result in [8] is the following:

**Theorem 6** ([8], Theorem 1)**.** *Let θ*<sup>0</sup> *be as in Proposition 5. Then for all θ* ∈ *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup> , Cn*ð Þ*<sup>θ</sup> is homeomorphic to S<sup>n</sup>*�<sup>4</sup>*.*

**Remark 7.** In [8], Theorem 6 is proved by the following method: We construct a function *f* : *Cn*ð Þ!*θ* and show that *f* has exactly two critical points. Then Reeb's

theorem implies Theorem 6. Note that with this method, we cannot improve the assertion from homeomorphism to diffeomorphism. (See ([10], p. 25) for Reeb's theorem and remarks about it.)

The following theorem is the answer to Problem 1 (i).

**Theorem A.** *We equip Sn*�<sup>4</sup> *with the standard differential structure. Let θ*<sup>0</sup> *be as in Proposition 5. Then for all θ* ∈ *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup> , Cn*ð Þ*<sup>θ</sup> is diffeomorphic to Sn*�4*.*

Next we consider Problem 1 (ii). We set

$$a\_{\mathfrak{n}} \coloneqq \inf \left\{ \theta\_0 \in (0, \pi) \, \middle| \, \mathbb{C}\_{\mathfrak{n}}(\theta) \cong \mathbb{S}^{n-4} \, \text{holds for all} \theta \in \left(\theta\_0, \frac{\mathfrak{n} - 2}{\mathfrak{n}}\pi\right) \right\}.$$

Here in what follows, the notation *X* ffi *Y* means that *X* is homeomorphic to *Y*. Note that among the values of *θ*<sup>0</sup> in Theorem 6, *α<sup>n</sup>* is the best possible one.

The following result is known.

**Theorem 8** ([11]). *(i) We have <sup>α</sup><sup>n</sup>* <sup>¼</sup> *<sup>n</sup>*�<sup>4</sup> *<sup>n</sup>*�<sup>2</sup> *<sup>π</sup> for* <sup>4</sup>≤*<sup>n</sup>* <sup>≤</sup>7*. (ii). We have α*<sup>8</sup> < <sup>5</sup> 7 *π.*

**Remark 9.** About Theorem 8 (i), we can read *α*<sup>4</sup> and *α*<sup>5</sup> from the above Example 4, and *α*<sup>6</sup> from **Table 4** in Theorem D below.

From Theorem 8, we naturally encounter the following:

**Question 10.** (i) Is it true that *<sup>α</sup><sup>n</sup>* <sup>¼</sup> *<sup>n</sup>*�<sup>4</sup> *<sup>n</sup>*�<sup>2</sup> *<sup>π</sup>* holds for *<sup>n</sup>*<sup>≥</sup> 4?

(ii) Is it true that *<sup>α</sup><sup>n</sup>* <sup>&</sup>lt; *<sup>n</sup>*�<sup>3</sup> *<sup>n</sup>*�<sup>1</sup> *<sup>π</sup>* holds for *<sup>n</sup>*≥4? Note that if (i) is true then (ii) holds automatically.

The following theorem is the answer to Problem 1 (ii).

**Theorem B.** *For* 4≤ *n*≤14*, the following* **Table 3** *holds.*

The following theorem is the answer to Question 10.

**Theorem C.** *(i) The statement in Question 10* (i) *is false for n*≥8*. In fact, we have n*�4 *<sup>n</sup>*�<sup>2</sup> *<sup>π</sup>* <sup>&</sup>lt; *<sup>α</sup><sup>n</sup> for n*<sup>≥</sup> 8.

*(ii) The statement in Question 10* (ii) *is false for n*<sup>≥</sup> <sup>13</sup>*. In fact, we have <sup>n</sup>*�<sup>3</sup> *<sup>n</sup>*�<sup>1</sup> *<sup>π</sup>* <sup>&</sup>lt;*α<sup>n</sup> for n*≥ 13*.*

The following theorem is the answer to Problem 1 (iii).

**Theorem D.** *The topological type of C*6ð Þ*θ is given by the following* **Table 4***, where we omit the cases which can be read from Theorem 3.*


**Table 3.** *The value of α<sup>n</sup> for* 4 ≤*n* ≤14*.*

*The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*


**Table 4.**

*The topological type of C*6ð Þ*θ .*

**Remark 11.** (i) As indicated in Problem 1 (iii), not only *S*<sup>2</sup> but also # 3 *<sup>S</sup>*<sup>1</sup> � *<sup>S</sup>*<sup>1</sup> � �

appears in **Table 4** as a topological type of *C*6ð Þ*θ*

(ii) We can also determine the topological type of *C*<sup>6</sup> *<sup>π</sup>* 2 � � and *C*<sup>6</sup> *<sup>π</sup>* 3 � �. (See Remark 18 in §5.) In particular, they have singular points.

#### **3. Proof of Theorem A**

Following the method of [12], we set

$$X\_n := \left\{ (P, \theta) \in \left( \mathbb{S}^2 \right)^n \times \left( 0, \frac{n-2}{n} \pi \right] \: \mid \: P \in \mathcal{C}\_n(\theta) \right\}. \tag{4}$$

We define the function *μ* : *Xn* ! by

$$
\mu(P, \theta) = \theta.\tag{5}
$$

Note that for all *θ* ∈ ð � 0, *π* , we have

$$
\mu^{-1}(\theta) = \mathsf{C}\_{\mathsf{n}}(\theta). \tag{6}
$$

The following proposition holds:

**Proposition 12.** (i) *Let θ*<sup>0</sup> *be as in Proposition 5. Then the space μ*�<sup>1</sup> *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � � *is a manifold, where μ*�<sup>1</sup> *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � � *denotes the inverse image of the interval.*

(ii) *Any element of μ*�<sup>1</sup> *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � � *is a regular point of <sup>μ</sup>.*

(iii) *Consider the case n* ¼ 8*. Then C*<sup>8</sup> 6 <sup>8</sup> *<sup>π</sup>* � � *is a non-degenerate critical point of <sup>μ</sup>.* In order to prove the proposition, we need a lemma. We set

$$D\_n := \left\{ (u\_1, \dots, u\_{n-2}, \theta) \in \left(\mathbb{S}^2\right)^{n-2} \times (0, \pi) \: \mid \text{ the following i andii hold} \right\}.$$

$$\text{i. } u\_1 = (1, 0, 0).$$

ii. *ui* h i , *ui*þ<sup>1</sup> ¼ � cos *θ* for 1≤ *i*≤ *n* � 3.

**Lemma 13.** *(i) There is a diffeomorphism*

$$f: \left(\mathbb{S}^1\right)^{n-3} \times (\mathbf{0}, \pi)\_{\underset{\rightleftarrows}{\rightrightarrows}} D\_n\dots$$

*(ii) We define the map L* : *Dn* ! *by*

$$L(u\_1, \dots, u\_{n-2}, \theta) = \left\|(-\cos\theta, -\sin\theta, \mathbf{0}) + \sum\_{i=1}^{n-2} u\_i\right\|^2.$$

*Then we have the following commutative diagram*:

$$\lambda\_{\mu} \xrightarrow[\lambda\_{\text{R}}]{\mathcal{S}} \searrow \mathcal{I} \tag{7}$$

*Here the maps p and g are defined as follows*.

• *Let*

$$Pr: \left(\mathbb{S}^1\right)^{n-3} \times (\mathbf{0}, \mathfrak{x}) \to \mathbb{R} \tag{8}$$

*be the projection to the* ð Þ 0, *π -component and we denote by p the restriction of Pr to* ð Þ *<sup>L</sup>* <sup>∘</sup>*<sup>f</sup>* �<sup>1</sup> ð Þ1 :

$$p := \Pr\vert\_{\left(L, \circ f\right)^{-1}(\mathbf{1})}.$$

• *The map g is a homeomorphism which will be defined in* (13).

*(iii) For all θ* ∈ð Þ 0, *π , the restriction of the map g in* (7) *naturally induces a homeomorphism*

$$\mathcal{g}|\_{\mathbb{C}\_{n}(\theta)} : \mathbb{C}\_{n}(\theta) \underset{\underline{\alpha}}{\to} p^{-1}(\theta). \tag{9}$$

**Proof of Lemma 13***:* (i) From an element

$$\left(e^{i\phi\_1}, \dots, e^{i\phi\_{n-3}}, \theta\right) \in \left(\mathbb{S}^1\right)^{n-3} \times (\mathbf{0}, \pi),$$

we construct the element ð Þ *u*1, ⋯, *un*�2, *θ* ∈ *Dn* as follows: In the process of constructing *ui*, we also construct the elements *vi* <sup>∈</sup>*S*<sup>2</sup> such that *ui* h i¼ , *vi* 0. We set

$$u\_{i+1} \coloneqq -\left(\cos\theta\right)u\_i + \left(\sin\theta\cos\phi\_i\right)v\_i + \left(\sin\theta\sin\phi\_i\right)u\_i \times v\_i \tag{10}$$

and

$$v\_{i+1} \coloneqq -\left(\sin\theta\right)u\_i - \left(\cos\theta\cos\phi\_i\right)v\_i - \left(\cos\theta\sin\phi\_i\right)u\_i \times v\_i,\tag{11}$$

where *ui* � *vi* denotes the cross product.

In (10) and (11) for *i* ¼ 1, we set *u*<sup>1</sup> ≔ ð Þ 1, 0, 0 and *v*<sup>1</sup> ≔ ð Þ 0, 1, 0 . Then we obtain *u*<sup>2</sup> and *v*2. Next using (10) and (11) for *i* ¼ 2, we obtain *u*<sup>3</sup> and *v*3. Repeating this process, we obtain *ui* and *vi* for 1≤*i* ≤*n* � 2. Now we define *f* by

$$f\left(e^{i\phi\_1}, \cdots, e^{i\phi\_{n-3}}, \theta\right) := (u\_1, \cdots, u\_{n-2}, \theta).$$

From the construction, *f* is a diffeomorphism.

(ii) We define the map *<sup>h</sup>* : *Xn* ! *<sup>L</sup>*�<sup>1</sup> ð Þ1 by

$$h(u\_1, \dots, u\_n, \theta) \coloneqq (u\_1, \dots, u\_{n-2}, \theta). \tag{12}$$

*The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*

Since ð Þ *u*1, ⋯, *un* is an element of *Cn*ð Þ*θ* , the right-hand side of (12) is certainly an element of *L*�<sup>1</sup> ð Þ1 . It is clear that *h* is a homeomorphism. Hence if we define the map *g* by

$$g \coloneqq f^{-1} \circ h,\tag{13}$$

then *g* is also a homeomorphism. From the construction, it is clear that the diagram (7) is commutative.

(iii) The item is clear from (6) and the diagram (7).

**Proof of Proposition 12:** Recall that ð Þ *<sup>L</sup>* <sup>∘</sup>*<sup>f</sup>* �<sup>1</sup> ð Þ1 in (7) is a subspace of

*S*<sup>1</sup> � �*n*�<sup>3</sup> � ð Þ 0, *π* . In order to prove Proposition 12, we calculate in the universal covering space. Let

$$q: \mathbb{R}^{n-3} \times (\mathbf{0}, \pi) \to \left(\mathbb{S}^1\right)^{n-3} \times (\mathbf{0}, \pi)$$

be the universal covering space and we define the map

$$
\widetilde{Pr} : \mathbb{R}^{n-3} \times (\mathbf{0}, \pi) \to \mathbb{R}
$$

by *Pr*e ≔ *Pr*∘ *q*, where the map *Pr* is defined in (8). Then in addition to (7), we have the following commutative diagram:

(i) Let *<sup>x</sup>*<sup>∈</sup> *<sup>n</sup>*�<sup>3</sup> � ð Þ 0, *<sup>π</sup>* be any element which satisfies the condition

$$q(\varkappa) \in p^{-1}\left(\theta\_0, \frac{n-2}{n}\pi\right].\tag{15}$$

Note that if we use the diagram (14), then (15) is equivalent to saying that

$$
\widetilde{Pr}(\mathbf{x}) \in \left(\theta\_0, \frac{n-2}{n}\pi\right] \quad \text{and} \quad (L \circ f \circ q)(\mathbf{x}) = \mathbf{1}.
$$

We set

$$\operatorname{grad}\_{\boldsymbol{x}}(\boldsymbol{L}\circ\boldsymbol{f}\circ\boldsymbol{q}) \coloneqq \left(\frac{\partial(\boldsymbol{L}\circ\boldsymbol{f}\circ\boldsymbol{q})}{\partial\phi\_{1}}(\boldsymbol{\omega}),\ldots,\frac{\partial(\boldsymbol{L}\circ\boldsymbol{f}\circ\boldsymbol{q})}{\partial\phi\_{n-3}}(\boldsymbol{\omega}),\frac{\partial(\boldsymbol{L}\circ\boldsymbol{f}\circ\boldsymbol{q})}{\partial\theta}(\boldsymbol{\omega})\right). \tag{16}$$

In order to prove Proposition 12 (i), it will suffice to prove that

$$\operatorname{grad}\_{\mathfrak{x}}(L\circ f \circ q) \neq (0, \cdots, 0) \tag{17}$$

(a) *The case when Pr x* <sup>e</sup> ð Þ¼ *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> π.*

We claim that *x* has the form

$$\boldsymbol{\omega} = \left( \mathbf{0}, \cdots, \mathbf{0}, \frac{n-2}{n}\boldsymbol{\pi} \right). \tag{18}$$

To prove this, recall the homeomorphism *g*j *Cn*ð Þ*<sup>θ</sup>* was defined in (9). Since *g*j *Cn*ð Þ*θ* � ��<sup>1</sup> ð Þ *q x*ð Þ is the regular *n*-gon, (18) follows.

We shall prove that

$$\mathbf{grad}\_{\mathbf{x}}(L\circ f \circ q) = (0, \cdots, 0, r) \tag{19}$$

for some positive real number *r*.

First, note that the real-valued function ð Þ *<sup>L</sup>* <sup>∘</sup>*<sup>f</sup>* <sup>∘</sup> *<sup>q</sup> <sup>ϕ</sup>*1, <sup>⋯</sup>, *<sup>ϕ</sup>n*�3, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � � takes the minimum value 0 at *ϕ*1, ⋯, *ϕn*�<sup>3</sup> ð Þ¼ ð Þ 0, ⋯, 0 . Hence the first ð Þ *n* � 3 -terms of the both sides of (19) coincide.

Second, direct computations show that

$$\begin{cases} (L \circ f \circ q)(0, \dots, 0, \theta) = \\ \begin{cases} 4\left(\sum\_{i=1}^{m} (-1)^{i} \sin\frac{2i-1}{2}\theta\right)^{2}, & \text{if } n = 2m+1, \\ \left(1 + 2\sum\_{i=1}^{m-1} (-1)^{i} \cos i\theta\right)^{2}, & \text{if } n = 2m. \end{cases} \end{cases} \tag{20}$$

The number *<sup>r</sup>* in (19) equals to the derivative of (20) at *<sup>θ</sup>* <sup>¼</sup> *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> π*. It is easy to see that

$$\begin{cases} (-1)^i \cos \frac{(2i-1)(2m-1)}{4m+2} \pi < 0, & \text{for } 1 \le i \le m, \\\ (-1)^{i+1} \sin \frac{i(2m-2)}{2m} \pi > 0, & \text{for } 1 \le i \le m-1 \end{cases} \tag{21}$$

and

$$\begin{cases} \sum\_{i=1}^{m} (-1)^{i} \sin \frac{(2i-1)(2m-1)}{4m+2} \pi = -\frac{1}{2}, \\\ 1 + 2 \sum\_{i=1}^{m-1} (-1)^{i} \cos \frac{i(2m-2)}{2m} \pi = 1. \end{cases} \tag{22}$$

Using (21) and (22), we can check that the derivative of (20) at *<sup>θ</sup>* <sup>¼</sup> *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> π* is positive, i.e., *r* is positive. Thus we have obtained (19). This completes the proof of (17) for the case (a).

(b) *The case when Pr x* <sup>e</sup> ð Þ<sup>∈</sup> *<sup>θ</sup>*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � �*.*

By Proposition 5, we have

$$\left(\frac{\partial (L \circ f \circ q)}{\partial \phi\_1}(\boldsymbol{x}), \dots, \frac{\partial (L \circ f \circ q)}{\partial \phi\_{n-3}}(\boldsymbol{x})\right) \neq (\mathbf{0}, \dots, \mathbf{0}).\tag{23}$$

Then using (16), we obtain (17). This completes the proof of (17) for the case (b), and hence also that of (i).

(ii) In order to prove by contradiction, assume that *μ*�<sup>1</sup> *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � � contains a critical point of *μ*. Then using (7), *p*�<sup>1</sup> *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � � contains a critical point of *<sup>p</sup>*. Lifting to the universal covering space using (14), there exists an element *<sup>x</sup>*<sup>∈</sup> *<sup>n</sup>*�<sup>3</sup> � ð Þ 0, *<sup>π</sup>* which satisfies the following two items:

• We have *Pr x* <sup>e</sup> ð Þ<sup>∈</sup> *<sup>θ</sup>*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � �. *The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*

• The point *x* is a critical point of the function *Pr*e under the constraint

$$(L \circ f \circ q)(\phi\_1, \cdots, \phi\_{n-3}, \theta) = 1. \tag{24}$$

We apply the Lagrange multiplier method to (24). Since *Pr*e *ϕ*1, ⋯, *ϕn*�<sup>3</sup> ð , *θ*Þ ¼ *θ*, there exists *λ*∈ such that

$$(0, \cdots, 0, 1) = \lambda \operatorname{grad}\_{\mathfrak{x}} (L \circ f \circ q). \tag{25}$$

We compare the first ð Þ *n* � 3 -components of the both sides of (25). Then by (23), we have *λ* ¼ 0. But this contradicts the last component of (25). Hence (25) cannot occur. This completes the proof of (ii).

(iii) Consider the Eq. (24). Using the implicit function theorem, we may assume that *θ* is a function with variables *ϕ<sup>i</sup>* ð Þ 1≤*i*≤ *n* � 3 : *θ* ¼ *θ ϕ*1, ⋯, *ϕn*�<sup>3</sup> ð Þ. Note that

$$Pr(\phi\_1, \dots, \phi\_{n-3}, \theta(\phi\_1, \dots, \phi\_{n-3})) = \theta(\phi\_1, \dots, \phi\_{n-3})\dots$$

Hence it will suffice to prove the following result for *n* ¼ 8:

$$\left| \left( \frac{\partial^2 \theta(\phi\_1, \cdots, \phi\_{n-3})}{\partial \phi\_i \partial \phi\_j} (\mathbf{0}, \cdots, \mathbf{0}) \right)\_{1 \le i, j \le n-3} \right| \neq \mathbf{0},\tag{26}$$

where jj denotes the determinant. Computing by the method of second implicit derivative, we see that the value of the left-hand side of (26) for *<sup>n</sup>* <sup>¼</sup> 8 is <sup>1</sup>� ffiffi 2 p <sup>16384</sup>. Hence (26) holds for *n* ¼ 8. This completes the proof of (iii), and hence also that of Proposition 12.

In order to prove Theorem A, we recall the following:

**Theorem 14** ([13], Corollary B). *For d*≥2*, let M be a d-dimensional smooth manifold without boundary and F* : *M* ! *a smooth function. We set* max *F M*ð Þ¼ *m and assume that m is attained by unique point z*∈ *M. Let a*∈ *satisfy the following four conditions:*

i. *a*< *m*.


*Then there is a diffeomorphism F*�<sup>1</sup> ð Þffi *<sup>a</sup> <sup>S</sup><sup>d</sup>*�<sup>1</sup> . **Remark 15.** For the proof of Theorem 14, the *h*-cobordism theorem (see [14],

p. 108, Proposition A) is crucial. Hence we cannot drop the condition *d* 6¼ 5.

**Proof of Theorem A:** First, we consider the case *n* 6¼ 8.

• For *F* in Theorem 14, we consider

$$\mu: \mu^{-1}\left(\theta\_0, \frac{n-2}{n}\pi\right] \to \mathbb{R}.$$

More precisely, we denote the restriction of *μ* in (5) to *μ*�<sup>1</sup> *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � � by the same symbol *μ*. Note that from the definition of *F*, *z* in Theorem 14 is *Cn <sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* � �.


Below we check that the conditions i, ii, iii and iv in Theorem 14 are satisfied. The items i and ii are clear. The item iii follows from Proposition 12 ii. The item iv follows from the following argument: Since dim *Xn* ¼ *n* � 3, we have dim *μ*�<sup>1</sup> *θ*0, *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> <sup>π</sup>* 6¼ 5 if and only if *<sup>n</sup>* 6¼ 8.

Now we can apply Theorem 14 and obtain that *<sup>μ</sup>*�1ð Þ*<sup>θ</sup>* is diffeomorphic to *Sn*�<sup>4</sup> if *θ* satisfies that *θ*<sup>0</sup> <*θ* < *<sup>n</sup>*�<sup>2</sup> *<sup>n</sup> π*. By (6), this is equivalent to saying that *Cn*ð Þ*θ* is also diffeomorphic to *<sup>S</sup>n*�4. This completes the proof of Theorem A for *<sup>n</sup>* 6¼ 8.

Second, we consider the case *n* ¼ 8. If we apply the Morse lemma to Proposition 12 (iii), then we obtain that *<sup>μ</sup>*�1ð Þ*<sup>θ</sup>* is diffeomorphic to *<sup>S</sup>*<sup>4</sup> if *<sup>θ</sup>* satisfies that *θ*<sup>0</sup> <*θ* < <sup>6</sup> <sup>8</sup> *<sup>π</sup>*. Hence *<sup>C</sup>*8ð Þ*<sup>θ</sup>* is also diffeomorphic to *<sup>S</sup>*4. This completes the proof of Theorem A for *n* ¼ 8.

#### **4. Proofs of Theorems B and C**

**Proof of Theorem B:** In ([8], Lemma 1), certain conditions on an element ð Þ *u*1, ⋯, *un* of *Cn*ð Þ*θ* are listed. For example, a condition is given by

$$
u\_2 = 
u\_n.\tag{27}$$

Let Λ be the set of the conditions. For *λ*∈Λ, we set

$$\Theta\_{\lambda} \coloneqq \inf \quad \{ \theta\_0 \in \left( 0, \frac{n-2}{n} \pi \right) \: \mid \text{ for all } \theta \in \left( \theta\_0, \frac{n-2}{n} \pi \right), \mathbb{C}\_n(\theta) \text{ does not } \\ \tag{28}$$
 
$$\text{continuan element which satisfies the condition } \lambda$$

containan element which satisfies the condition*λ*g*:*

Using this, we set

$$
\beta\_n \coloneqq \max \quad \{\Theta\_{\vec{\lambda}} \: \mid \: \vec{\lambda} \in \Lambda\}. \tag{29}
$$

Then it is proved in ([8], Proposition 1) that

$$a\_n = \beta\_n.\tag{30}$$

We explain how to compute Θ*λ*. As an example of *λ*, we consider the condition (27). We construct the continuous function

$$R\_{\lambda} : (\mathbf{0}, \boldsymbol{\pi}) \to \mathbb{R} \tag{31}$$

which satisfies the following two properties:


In order to construct *R<sup>λ</sup>* in (31), we first fix *θ* and define the space *Yn*ð Þ*θ* as follows:

*The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*

$$Y\_n(\theta) \coloneqq \left\{ (u\_1, \dots, u\_n) \in \left(\mathbb{S}^2\right)^n \mid \text{ the following i and ii hold} \right\}.$$

$$\text{i. } u\_1 = (\mathbf{1}, \mathbf{0}, \mathbf{0}) \text{ and } u\_2 = u\_n = (-\cos\theta, -\sin\theta, \mathbf{0}).$$

$$\text{iii. } \langle u\_i, u\_{i+1} \rangle = -\cos \theta \text{ for } 2 \le i \le n-3.$$

Second, we define the function *r<sup>λ</sup>* : *Yn*ð Þ! *θ* as follows: For ð Þ *u*1, ⋯, *un* ∈ *Yn*ð Þ*θ* , we set

$$r\_{\lambda}(u\_1, \dots, u\_n) := \left\| \sum\_{i=1}^n u\_i \right\|. \tag{32}$$

Third, we define *R<sup>λ</sup>* in (31) by

$$R\_{\lambda}(\theta) \coloneqq \min r\_{\lambda}(Y\_n(\theta)).$$

Below we check the above properties a and b of *Rλ*.

The item a is clear.

In order to prove the item b, we claim the following identification holds:

$$r\_{\boldsymbol{\lambda}}^{-1}(\mathbf{0}) = \{ (\boldsymbol{u}\_1, \dots, \boldsymbol{u}\_n) \in \mathcal{C}\_n(\boldsymbol{\theta}) \: \mid \: \boldsymbol{u}\_2 = \boldsymbol{u}\_n \}. \tag{33}$$

In fact, an element ð Þ *<sup>u</sup>*1, <sup>⋯</sup>, *un* <sup>∈</sup>*Yn*ð Þ*<sup>θ</sup>* belongs to *<sup>r</sup>*�<sup>1</sup> *<sup>λ</sup>* ð Þ 0 if and only if (1) ii holds. Hence (33) follows.

Now the item b is clear from (33). Thus we have checked the above properties a and b.

Next using the properties a and b, we can describe Θ*<sup>λ</sup>* in (28) as

$$\Theta\_{\vec{\lambda}} = \max \left\{ \theta \in (\mathbf{0}, \pi) \: \mid \: R\_{\vec{\lambda}}(\theta) = \mathbf{0} \right\}. \tag{34}$$

From the constructions in (10) and (11), we have

$$Y\_n(\theta) \cong \left(\mathbb{S}^1\right)^{n-4} \times \mathbb{S}^2.$$

Using this fact, we can compute the right-hand side of (34) for *n*≤ 14.

By a similar method, we compute Θ*<sup>λ</sup>* for each *λ*∈Λ. Then from the definition of *β<sup>n</sup>* in (29), we can determine *βn*. Finally, using (30), we obtain *αn*. This completes the proof of Theorem B.

**Remark 16.** In the above proof of Theorem B, the identification (33) is crucial. Although *r*�<sup>1</sup> *<sup>λ</sup>* ð Þ 0 is a critical submanifold of the function *r<sup>λ</sup>* in (32), this fact allows us to compute the right-hand side of (34) for *n* ≤14. See §6 (ii) for further remarks.

**Proof of Theorem C:** The theorem is clear from **Table 3**.

#### **5. Proof of Theorem D**

The following proposition is a refinement of Proposition 12 for *n* ¼ 6. **Proposition 17.** *(i) The space X*<sup>6</sup> *is a manifold, where Xn is defined in* (4). *(ii) The interior angle θ is a critical point of μ if and only if θ equals to <sup>π</sup>* 3*, π* <sup>2</sup> *or* <sup>2</sup> 3 *π.*

**Proof:** We can prove the proposition is the same way as in Proposition 12. Since the dimension is low, we can perform direct computations.

#### *Advanced Topics of Topology*

We apply the first fundamental theorem of Morse theory to Proposition 17. Then we obtain the following assertion: If *θ*<sup>1</sup> and *θ*<sup>2</sup> belong to the same interval from the three intervals 0, *<sup>π</sup>* 3 � �, *<sup>π</sup>* <sup>3</sup> , *<sup>π</sup>* 2 � � and *<sup>π</sup>* 2 , 2 <sup>3</sup> *<sup>π</sup>* � �, then we have *<sup>C</sup>*6ð Þffi *<sup>θ</sup>*<sup>1</sup> *<sup>C</sup>*6ð Þ *<sup>θ</sup>*<sup>2</sup> .

The homeomorphism (9) tells us that in order to determine the topological type of *<sup>C</sup>*6ð Þ*<sup>θ</sup>* , it will suffice to determine the topological type of *<sup>p</sup>*�1ð Þ*<sup>θ</sup>* for *<sup>n</sup>* <sup>¼</sup> 6. For a fixed *ψ* ∈½ � 0, 2*π* , we set

$$\mathcal{M}\_{\theta}(\boldsymbol{\psi}) := \left\{ \left( \boldsymbol{e}^{i\phi\_1}, \boldsymbol{e}^{i\phi\_2}, \boldsymbol{e}^{i\phi\_3}, \theta \right) \in \boldsymbol{p}^{-1}(\theta) \; | \; \phi\_1 = \boldsymbol{\psi} \right\}.$$

Since *Mθ*ð Þ *ψ* is a one dimensional object, it is not to difficult to draw its figure. The results are given as follows.

(i) *The case when <sup>π</sup>* <sup>2</sup> <*θ* < <sup>2</sup> 3 *π.*

There exists *ω* in 0, ð Þ *π* such that the following homeomorphism holds:

$$M\_{\theta}(\boldsymbol{\psi}) \cong \begin{cases} \{\text{one point}\}, & \text{if } \boldsymbol{\psi} = \boldsymbol{\alpha} \text{ or } 2\pi - \boldsymbol{\alpha}, \\ \mathcal{Q}, & \text{if } \boldsymbol{\alpha} < \boldsymbol{\psi} < 2\pi - \boldsymbol{\alpha}, \\ \mathcal{S}^{1}, & \text{otherwise}. \end{cases}$$

From this, we have *<sup>p</sup>*�<sup>1</sup>ð Þffi *<sup>θ</sup> <sup>S</sup>*<sup>2</sup> . The figure of *C*6ð Þ*θ* is given by the following **Figure 6**.

(ii) *The case when <sup>π</sup>* <sup>3</sup> <sup>&</sup>lt;*<sup>θ</sup>* <sup>&</sup>lt; *<sup>π</sup>* 2*.*

There exists *ω* in 0, ð Þ *π* such that the following homeomorphism holds:

$$M\_{\theta}(\boldsymbol{\psi}) \cong \begin{cases} \sigma, & \text{if } \boldsymbol{\psi} = \boldsymbol{\alpha} \text{ or } 2\pi - \boldsymbol{\alpha}. \\ \mathbb{S}^{1}, & \text{otherwise}, \end{cases} \tag{35}$$

Here we set

$$\sigma := \bigcup\_{i=1}^{2} \left\{ (\mathbf{x}, \boldsymbol{\mathcal{y}}) \in \mathbb{R}^2 \; | \; \mathbf{x}^2 + i^2 \boldsymbol{\mathcal{y}}^2 = \mathbf{1} \right\}.$$

(The figure of *σ* is given by the following **Figure 5(b)**.)

We claim that the four intersection points in *Mθ*ð Þ *ω* ∪ *Mθ*ð Þ 2*π* � *ω* are saddle points of *<sup>p</sup>*�<sup>1</sup>ð Þ*<sup>θ</sup>* . In fact, for a sufficiently small positive real number *<sup>ε</sup>*, the following **Figure 5 (a)–(c)** give the shape of *Mθ*ð Þ *ω* � *ε* , *Mθ*ð Þ *ω* and *Mθ*ð Þ *ω* þ *ε* , respectively. (The deformation of the shape of *Mθ*ð Þ *ψ* when *ψ* is near 2*π* � *ω* is also given by **Figure 5**.) Now from **Figure 5**, we see that the four intersection points are in fact saddle points.

Since we identify *Mθ*ð Þ 0 with *Mθ*ð Þ 2*π* , (35) and **Figure 5** give the homeomorphism *<sup>p</sup>*�<sup>1</sup>ð Þffi *<sup>θ</sup>* # 3 *<sup>S</sup>*<sup>1</sup> � *<sup>S</sup>*<sup>1</sup> � �. The figure of *<sup>C</sup>*6ð Þ*<sup>θ</sup>* is given by the following **Figure 7**.

(iii) *The case when* 0< *θ* < *<sup>π</sup>* 3*.*

The topological type of *Mθ*ð Þ *ψ* is the same as (35). Hence the argument in (ii) remains valid.

**Remark 18.** We determine the topological type of *C*<sup>6</sup> *<sup>π</sup>* 2 � � and *C*<sup>6</sup> *<sup>π</sup>* 3 � �.

(i) The figure of *C*<sup>6</sup> *<sup>π</sup>* 2 � � is given by the following **Figure 8**.

**Figure 5.** *(a) Mθ*ð Þ *ω* � *ε ; (b) Mθ*ð Þ *ω ; (c) Mθ*ð Þ *ω* þ *ε .* *The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*

Thus *C*<sup>6</sup> *<sup>π</sup>* 2 is homeomorphic to the orbit space *S*<sup>2</sup> *=* �, where the equivalence relation is generated by

$$(-1,0,0) \sim (1,0,0), \quad (0,-1,0) \sim (0,1,0) \quad \text{and} \quad (0,0,-1) \sim (0,0,1).$$

In particular, *C*<sup>6</sup> *<sup>π</sup>* 2 has three singular pints.

As *θ* approaches *<sup>π</sup>* <sup>2</sup> from below, each center of the three handles in **Figure 7** shrinks. And when *<sup>θ</sup>* <sup>¼</sup> *<sup>π</sup>* 2 , each center pinches to a point and we obtain **Figure 8**. If *θ* increases further from *<sup>π</sup>* <sup>2</sup>, then the pinched point separates and we obtain **Figure 6**.

(ii) The figure of *C*<sup>6</sup> *<sup>π</sup>* 3 is given by the following **Figure 9**.

The space *C*<sup>6</sup> *<sup>π</sup>* 3 contains subspaces

*i*¼1

*i* <*j*

$$N\_1, \ N\_2 \text{ and } N\_3 \tag{36}$$

which satisfy the following three properties:

• *<sup>N</sup>*<sup>1</sup> ffi *<sup>S</sup>*<sup>1</sup> � *<sup>S</sup>*<sup>1</sup> , *:N*<sup>2</sup> ffi *<sup>S</sup>*<sup>1</sup> � *<sup>S</sup>*<sup>1</sup> and *<sup>N</sup>*<sup>3</sup> ffi # 2 *<sup>S</sup>*<sup>1</sup> � *<sup>S</sup>*<sup>1</sup> • ⋃ 3 *i*¼1 *Ni* <sup>¼</sup> *<sup>C</sup>*<sup>6</sup> *<sup>π</sup>* 3 *:* • ⋃ *Ni* ∩ *N <sup>j</sup>* ffi <sup>⋃</sup> 3 ð Þ *<sup>x</sup>*, *<sup>y</sup>* <sup>∈</sup> <sup>2</sup> <sup>j</sup> *<sup>x</sup>*<sup>2</sup> <sup>þ</sup> *<sup>i</sup>* <sup>2</sup> *<sup>y</sup>*<sup>2</sup> <sup>¼</sup> <sup>1</sup> *:*

**Figure 6.** *The space C*6ð Þ*<sup>θ</sup> for <sup>π</sup>* <sup>2</sup> <*θ* < <sup>2</sup> <sup>3</sup> *π.*

**Figure 7.** *The space C*6ð Þ*<sup>θ</sup> for <sup>π</sup>* <sup>3</sup> <sup>&</sup>lt;*<sup>θ</sup>* <sup>&</sup>lt; *<sup>π</sup>* <sup>2</sup>*, where we identify the opposite boundaries.*

**Figure 8.** *The space C*<sup>6</sup> *<sup>π</sup>* 2 *, where we identify the opposite vertices.*

*The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*

**Figure 9.** *The space C*<sup>6</sup> *<sup>π</sup>* 3 *.*

**Figure 10.** *The space C*<sup>6</sup> *<sup>π</sup>* <sup>3</sup> *ε .*

The figure of *C*<sup>6</sup> *<sup>π</sup>* <sup>3</sup> *<sup>ε</sup>* is given by **Figure 10** above.

As *θ* approaches *<sup>π</sup>* <sup>3</sup>, a cross-section of the four tubes in **Figure 7** becomes a union of two circles: In the notation of (36), one circle becomes a handle of *N*3. And the other circle is a subspace of *N*<sup>1</sup> ∪ *N*2.

On the other hand, the hole of the center of **Figure 7** becomes a subspace of *N*<sup>1</sup> ∪ *N*2.

**Figure 12.** *The space X*5*.*

*The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*

**Proof of Example 4:** We can prove the example in the same way as in Theorem D. We can also prove by the following method. Recall that the space *Xn* was defined in (4). The figures of *X*<sup>4</sup> and *X*<sup>5</sup> are given by **Figures 11** and **12** above, respectively.

The identification (6) tells us that each level set of **Figure 11** gives *C*4ð Þ*θ* , and that of **Figure 12** gives *C*5ð Þ*θ* . Thus we obtain Example 4.

#### **6. Conclusions**


$$\left(\frac{\partial (L \circ f \circ q)}{\partial \phi\_1}(\boldsymbol{x}), \dots, \frac{\partial (L \circ f \circ q)}{\partial \phi\_{n-3}}(\boldsymbol{x})\right) = (\mathbf{0}, \dots, \mathbf{0})\tag{37}$$

and

$$(L \circ f \circ q)(\phi\_1, \cdots, \phi\_{n-3}, \theta) = \mathbf{1}.\tag{38}$$

If we could solve the system of Eqs. (37) and (38) with respect to the variables *ϕ*1, ⋯, *ϕ<sup>n</sup>*�<sup>3</sup> and *θ*, then we could determine for which *θ*, *Cn*ð Þ*θ* has a singular point and the set of singular points of *Cn*ð Þ*θ* . In particular, we obtain Proposition 5. But it is not easy to solve a system of equations even if we can use a computer. Hence, as we remarked in Remark 16, we have given the proof of Theorem B such as in §4. We pose the following question: Is it possible to solve the system of Eqs. (37) and (38) with respect to the variables *ϕ*1, ⋯, *ϕ<sup>n</sup>*�<sup>3</sup> and *θ*?

#### **Author details**

Yasuhiko Kamiyama Faculty of Science, Department of Mathematical Sciences, University of the Ryukyus, Okinawa, Japan

\*Address all correspondence to: kamiyama@sci.u-ryukyu.ac.jp

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### **References**

[1] Kapovich M, Millson J. The symplectic geometry of polygons in the Euclidean space. Journal of Differential Geometry 1996; 44: 479–513. DOI: 10.4310/jdg/1214459218 https://projecte uclid.org/journals/journal-of-differentialgeometry/volume-44/issue-3/Thesymplectic-geometry-of-polygonsin-Euclidean-space/10.4310/jdg/ 1214459218.full [Accessed: 30 September 2021]

[2] Hausmann J.-C, Knutson A. The cohomology ring of polygon spaces. Annales de l'Institut Fourier 1998; 48; 281–321. DOI: 10.5802/aif.1619 https:// aif.centre-mersenne.org/item/AIF\_ 1998\_\_48\_1\_281\_0 [Accessed: 30 September 2021]

[3] Farber M. Invitation to Topological Robotics. Zurich Lectures in Advanced Mathematics: European Mathematical Society; 2008. 143p. DOI: 10.4171/054

[4] Crippen G. Exploring the conformation space of cycloalkanes by linearized embedding. Journal of Computational Chemistry 1992; 13: 351– 361. DOI:10.1002/jcc.540130308 https:// www.researchgate.net/publication/ 30839993\_Exploring\_the\_ conformation\_space\_of\_cycloalkanes\_ by\_ linearized\_embedding [Accessed: 30 September 2021]

[5] O'Hara J. The configuration space of equilateral and equiangular hexagons. Osaka Journal of Mathematics 2013; 50: 477–489. DOI:10.18910/25093 https:// projecteuclid.org/journals/osaka-journalof-mathematics/ volume-50/issue-2/Theconfiguration-space-of-equilateral-andequiangular- hexagons/ojm/1371833496. full [Accessed: 30 September 2021]

[6] Kamiyama Y. The configuration space of equilateral and equiangular heptagons. JP Journal of Geometry and Topology 2020, 25: 25–33 DOI:http://dx. doi.org/10.17654/GT025120025 https://

www.researchgate.net/publication/ 347482448\_THE\_CONFIGURATION\_ SPACE\_OF\_EQUILATERAL\_AND\_ EQUIANGULAR\_HEPTAGONS [Accessed: 30 September 2021]

[7] Goto S, Komatsu K. The configuration space of a mathematical model for ringed hydrocarbon molecules. Hiroshima Mathematical Journal 2012; 42: 115–126. DOI: 10.32917/hmj/1333113009 https://projec teuclid.org/journals/hiroshima-mathe matical-journal/volume-42/issue-1/Theconfiguration-space-of-a-model-forringed-hydrocarbon-molecules/ 10.32917/hmj/1333113009.full [Accessed: 30 September 2021]

[8] Goto S, Komatsu K, Yagi J. The configuration space of almost regular polygons. Hiroshima Mathematical Journal 2020; 50: 185–197. DOI: 10.32917/hmj/1595901626 https://projec teuclid.org/journals/hiroshima-mathe matical-journal/ volume-50/issue-2/The-configuration-space-of-almostregular-polygons/ 10.32917/hmj/ 1595901626.full [Accessed: 30 September 2021]

[9] Kamiyama Y. A filtration of the configuration space of spatial polygons. Advances and Applications in Discrete Mathematics 2019; 22: 67–74. DOI:http:// dx.doi.org/10.17654/DM022010067 https://www.researchgate.net/publica tion/ 336117647\_A\_FILTRATION\_OF\_ THE\_CONFIGURATION\_SPACE\_OF\_ SPATIAL\_POLYGONS [Accessed: 30 September 2021]

[10] Milnor J. Morse Theory. Annals of Mathematics Studies Volume 51: Princeton University Press; 1963. 153p. DOI: 10.1515/9781400881802

[11] Goto S, Hemmi Y, Komatsu K, Yagi J. The closed chains with spherical configuration spaces. Hiroshima Mathematical Journal 2012, 42: 253–266. *The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes DOI: http://dx.doi.org/10.5772/intechopen.100723*

DOI: 10.32917/hmj/1345467073 https:// projecteuclid.org/journals/hiroshimamathematical-journal/ volume-42/issue-2/The-closed-chains-with-sphericalconfiguration-spaces/ 10.32917/hmj/ 1345467073.full [Accessed: 30 September 2021]

[12] Kamiyama Y. The Euler characteristic of the regular spherical polygon spaces. Homology Homotopy and Applications 2020, 22: 1–10. DOI: https://dx.doi.org/10.4310/HHA.2020. v22.n1.a1 https://www.intlpress.com/ site/pub/pages/journals/items/hha/ content/vols/0022/0001/a001 [Accessed: 30 September 2021]

[13] Kamiyama Y. On the level set of a function with degenerate minimum point. International Journal of Mathematics and Mathematical Sciences 2015, Article ID 493217: 6pages. DOI:10.1155/2015/493217 https://downloads.hindawi.com/ journals/ijmms/2015/493217.pdf [Accessed: 30 September 2021]

[14] Milnor J. Lectures on the h-cobordism Theorem. Princeton University Press: Princeton; 1965. 122p. DOI: 10.1515/9781400878055

#### **Chapter 5**

## Covers and Properties of Families of Real Functions

*Lev Bukovský*

#### **Abstract**

We present results on the relationships of the covering property Gð Þ Φ, Ψ for Φ, Ψ ∈ f g O,Λ, Ω, Γ and G ∈ f g S1, Sfin, *U*fin of a topological space and the selection property Gð Þ Φ**0**, Ψ**<sup>0</sup>** of the corresponding family of real functions. The result already published are presented without a proof, however with a citation of the corresponding paper. We present a general Theorem that covers almost all the result of this kind. Some results about hereditary properties are enclosed. We also present Scheepers Diagram of considered covering properties for uncountable covers.

**Keywords:** covering properties S1, Sfin, *U*fin, selection principles S1, Sfin, *U* <sup>∗</sup> fin, Scheepers Diagram, A-measurable function, upper A-semimeasurable function, hereditary properties, *σ*-space

#### **1. Introduction**

The paper is a collection of several results concerning the equivalences of the covering properties of a topological space *X* and the properties of the family of real functions defined on *X* and related to this cover. Indeed, we shall present results about the equivalences of the covering properties Gð Þ Φ, Ψ of a topological space *X* and the selection property Gð Þ Φ**0**, Ψ**<sup>0</sup>** of the topological space of upper semicontinuous real functions USC*p*ð Þ *X* on *X*, for Φ, Ψ ∈f g O,Λ, Ω, Γ and G ∈ f g S1, Sfin, *U*fin . The upper semicontinuous functions in this connection were for first time used in [1]. In some important cases we can replace the space USC*p*ð Þ *X* by the topological space C*p*ð Þ *X* of continuous functions. So we obtain the equivalence of some topological property of C*p*ð Þ *X* and a covering property of *X*.

It turned out that we can prove a general theorem about measurable covers in a very abstract sense that covers almost all the special results.

The covering properties Gð Þ Φ, Ψ were essentially introduced by M. Scheepers [2] and then, for countable covers, systematically investigated by W. Just, A.W. Miller, M. Scheepers and P.J. Szeptycki [3]. Using proved equivalences they obtained the quite simple Scheepers Diagram for Countable Covers. However, not all equivalences are true for arbitrary covers, therefore the corresponding Scheepers Diagram for Arbitrary Infinite Covers, presented below, is more complicated.

Then we present some results about hereditary properties of considered covering properties for *Fσ*-subsets. The results are important in many considerations. Finally, inspired by the result of J. Haleš [4], we show some relationships of the

hereditary property of the topological space *X* for any subset with the property being a *σ*-space.

If the presented result was already published, we present the precise citation and no proof.

#### **2. Notations and terminology**

By **a topological space** we understand an infinite Hausdorff space h i *X*, *τ* , where *τ* is the topology on *X*: the set of all open subsets of *X* [5]. The smallest *σ*-algebra containing all open sets is the family BORELð Þ¼ *X* BOREL of Borel subsets of *X*.

We recall some covering properties which were introduced by M. Scheepers in [2]. If A,B ⊆Pð Þ *Y* are sets of subsets of a set *Y*, then the *covering principle* S1ð Þ A, B means the following: for every sequence h i U*<sup>n</sup>* : *n*∈ *ω* of elements of A and for every *n*∈ *ω* there exists a *Un* ∈U*<sup>n</sup>* such that f g *Un* : *n* ∈*ω* ∈ B. The covering principles Sfinð Þ A, B and *U*finð Þ A, B are there defined in a similar way. In the case of Sfinð Þ A, B we choose finite sets W*<sup>n</sup>* ⊆U*<sup>n</sup>* such that ⋃*n*W*<sup>n</sup>* ∈B. In the case of Sfinð Þ A, B we chose finite sets W*<sup>n</sup>* ⊆U*<sup>n</sup>* such that ⋃*n*W*<sup>n</sup>* ∈B. In the case of *U*finð Þ A, B we chose finite sets W*<sup>n</sup>* ⊆U*<sup>n</sup>* such that f g ⋃W*<sup>n</sup>* : *n*∈*ω* ∈ B. Actually we tacitly assume that *<sup>Y</sup>* <sup>¼</sup> <sup>P</sup>ð Þ *<sup>X</sup>* . If *<sup>Y</sup>* <sup>⊆</sup>*<sup>X</sup>* we define the selection property *<sup>U</sup>* <sup>∗</sup> finð Þ A, B similarly, but we ask<sup>1</sup> that min f g <sup>W</sup>*<sup>n</sup>* : *<sup>n</sup>*<sup>∈</sup> *<sup>ω</sup>* <sup>∈</sup>B.

A family U of subsets of *X* is a *cover* of *X* if ⋃U ¼ *X*. A cover is *open*, if every element of the cover is an open set. A cover V ⊆U is said to be a *subcover* of the cover U. If we deal with a countable cover of *X* we can consider it as a sequence of subsets.

If A is a family of subsets of *X*, then we denote by Actbl the family of all countable elements of <sup>A</sup>. If <sup>A</sup> <sup>⊆</sup>*<sup>X</sup>* then <sup>A</sup><sup>þ</sup> is the family f g *<sup>f</sup>* <sup>∈</sup> <sup>A</sup> : ð Þ <sup>∀</sup>*x*∈*<sup>X</sup> f x*ð Þ<sup>≥</sup> <sup>0</sup> .

We introduce four special types of covers of a given set. From some technical reasons a cover is said to be an *o*-*cover*. A cover U of a set *X* is a *λ*-*cover* if for every *x*∈*X* the set f g *U* ∈ U : *x*∈ *U* is infinite. A cover U of a set *X* is an *ω*-*cover* if *X* ∉ U and for every finite *F* ⊆*X* there exists a *U* ∈ U such that *F* ⊆ *U*. Finally, a cover U of a set *X* is a *γ*-*cover* if U is infinite and for every *x*∈*X* the set f g *U* ∈U : *x* ∉ *U* is finite.

We shall use the following convention. If the lower case letters *φ* or *ψ* denote one of the symbols *o*, *λ*, *ω* or *γ*, then the capital letters Φ or Ψ denote the corresponding symbol O, Λ, Ω or Γ, respectively, and vice versa.

Assume that E ⊆Pð Þ *X* and ∅, *X* ∈E. Dealing with the covering property *U*fin we assume that E is closed under finite unions.

Let *φ*∈ f g *o*, *λ*, *ω*, *γ* . We denote by Φð Þ E the family of all *φ*-covers U of *X* satisfying U ⊆E. If h i *X*, *τ* is an infinite Hausdorff topological space, then Φð Þ*τ* is simply denoted as Φ.

Evidently (we should eventually omit *X* from a *γ*-cover)

$$
\Gamma(\mathcal{E}) \subseteq \Omega(\mathcal{E}) \subseteq \Lambda(\mathcal{E}) \subseteq \mathcal{O}(\mathcal{E}).\tag{1}
$$

We say that a family V ⊆Pð Þ *X* is a *refinement* of the family U ⊆Pð Þ *X* if

$$(\forall V \in \mathcal{V})(\exists U \in \mathcal{U}) \, V \subseteq U. \tag{2}$$

Let the family V ⊆Pð Þ *X* be a refinement of the family U ⊆Pð Þ *X* . If V is an *o*- or an *ω*-cover, then U is such a cover as well. This is not true for *λ*- and *γ*-covers. If we

<sup>1</sup> If W is finite then min W ¼ *h*, where *h x*ð Þ¼ min f g *f x*ð Þ : *f* ∈ W for every *x*∈*X*. min ∅ ¼ 1.

*Covers and Properties of Families of Real Functions DOI: http://dx.doi.org/10.5772/intechopen.100555*

add finitely many subsets of *X* to a *φ*-cover, *φ*∈f g *o*, *λ*, *ω*, *γ* , we obtain a *φ*-cover. Any infinite subset of a *γ*-cover is a *γ*-cover as well.

A set *<sup>ω</sup>* of all sequences of reals is quasi-ordered by the **eventual dominating** relation

$$
\varphi \le^\* \psi \equiv (\exists n\_0 \in \alpha) (\forall n \ge n\_0) \,\varphi(n) \le \psi(n). \tag{3}
$$

A set F ⊆*<sup>ω</sup>* is called **bounded** if there exists a sequence *ψ* ∈*<sup>ω</sup>* such that *φ*≤ <sup>∗</sup> *ψ* for every *φ* ∈ F. The set F is **dominating** if for every *ψ* ∈*<sup>ω</sup>* there exists a *φ*∈ F such that *ψ* ≤ <sup>∗</sup> *φ*. The cardinals b and d (see e.g. [6, 7]) are the smallest cardinalities of an unbounded and dominating family, respectively.

The set *<sup>X</sup>* is endowed with the product topology. For any real *a*∈ we denote by **a** the constant function defined on *X* with value *a*. There are at least two important subfamilies of *<sup>X</sup>*: the family *C X*ð Þ of all continuous functions and the family USCð Þ *<sup>X</sup>* of all upper semicontinuous functions<sup>2</sup> . If they are endowed with the product topology we write C*p*ð Þ *X* and USC*p*ð Þ *X* , respectively. Note that *f <sup>n</sup>* ! *f* in the product topology if and only if ð Þ ∀*x*∈*X f <sup>n</sup>*ð Þ! *x f x*ð Þ.

We introduce three properties of an infinite family *F* ⊆ *<sup>X</sup>* of real functions.


Let Φ and Ψ be one of the symbols O, Ω, Γ. Similarly as for covers, we define for an infinite family *F* ⊆*<sup>X</sup>* of real functions the set

$$\Phi\_{\mathbf{0}}(F) = \{ H \subseteq F : H \text{ is infinite and has the property } (\Phi\_{\mathbf{0}}) \}. \tag{4}$$

*F* **satisfies the selection principle** S1ð Þ Φ0, Ψ<sup>0</sup> if S1ð Þ Φ0ð Þ *F* , Ψ0ð Þ *F* holds. Identifying the countable sets of functions with sequences of functions, we say that *F* **satisfies the sequence selection principle** <sup>S</sup>1ð Þ <sup>Φ</sup>0, <sup>Ψ</sup><sup>0</sup> if <sup>S</sup><sup>1</sup> ð Þ <sup>Φ</sup>0ð Þ *<sup>F</sup>* ctbl,ð Þ <sup>Ψ</sup>0ð Þ *<sup>F</sup>* ctbl � � holds (compare, e.g., [8]: for each sequence of sequences *f <sup>n</sup>*,*<sup>m</sup>* : *m* ∈*ω* D E : *<sup>n</sup>* <sup>∈</sup>*<sup>ω</sup>* D E of functions from *<sup>F</sup>* with the property ð Þ <sup>Φ</sup>**<sup>0</sup>** , there exists a functions *<sup>α</sup>*∈*ωω* such that *<sup>f</sup> <sup>n</sup>*,*α*ð Þ *<sup>n</sup>* : *<sup>n</sup>* <sup>∈</sup>*<sup>ω</sup>* n o has the property ð Þ <sup>Ψ</sup>**<sup>0</sup>** .) Similarly for Sfin and *<sup>U</sup>*fin.

#### **3. Some results about the relationship of properties of covers and of families of real functions**

The first results about covering properties and the properties of the family of continuous real functions were obtained by W. Hurewicz [9]. He proved the following two theorems.

**Theorem 1** (W. Hurewicz [9])**.** *If X is a perfectly normal topological space then the following are equivalent.*

a. *X is a U*fin Octbl ð Þ , O

<sup>2</sup> A function *<sup>f</sup>* <sup>∈</sup>*<sup>X</sup>*<sup>ℝ</sup> is **upper semicontinuous** if the set f g *<sup>x</sup>*∈*<sup>X</sup>* : *f x*ð Þ<*<sup>a</sup>* is open for every real *<sup>a</sup>* <sup>∈</sup> <sup>ℝ</sup>.

b. *For every sequence f <sup>n</sup>* : *<sup>n</sup>*<sup>∈</sup> *<sup>ω</sup> of continuous real functions the family <sup>f</sup> <sup>n</sup>*ð Þ *<sup>x</sup>* : *<sup>n</sup>*<sup>∈</sup> *<sup>ω</sup>* : *<sup>x</sup>*∈*<sup>X</sup>* <sup>⊆</sup> *<sup>ω</sup> is bounded.*

**Theorem 2** (W. Hurewicz [9])**.** *If X is a perfectly normal topological space then the following are equivalent.*


Note that the property *U*fin Octbl ð Þ , Γ of a topological space was introduced and investigated by K. Menger [10].

Proofs of both Theorems may be found, e.g., in L. Bukovský and J. Haleš [11]. A topological space *X* is a *γ*-**space** if every open *ω*-cover of *X* has a countable *γ*subcover.

F. Gerlits and Z. Nagy [12] proved.

**Theorem 3** (F. Gerlits and Z. Nagy [12])**.** *If X is a Tychonoff topological space then the following are equivalent:*

a. C*p*ð Þ *X is Fréchet.*

b. *X is a γ-space.*

c. *X is an* S1ð Þ Ω, Γ *-space.*

A topological space *X* has **countable strong fan tightness** if *An* ⊆*X* and *x*∈ *An*, *n*∈ *ω* imply that there exists a sequence h i *xn* : *n*∈ *ω* such that *xn* ∈ *An* and *x*∈f g *xn* : *n*∈ *ω* .

**Theorem 4** (M. Sakai [13])**.** *A Tychonoff topological space has the covering property* S1ð Þ Ω, Ω *if and only if the topological space* C*p*ð Þ *X has countable strong fan tightness.*

That was M. Scheepers [2] who began the systematic study of the covering properties Gð Þ Φ, Ψ for G ¼ S1, Sfin, *U*fin, Φ, Ψ ¼ O,Λ, Ω, Γ.

The first use of upper semicontinuous functions in the study of covering properties was.

**Theorem 5** (L. Bukovský [1])**.** *A topological space X is an* S1ð Þ Γ, Γ *-space if and only if* USC*p*ð Þ *X* <sup>þ</sup> *satisfies the selection principle* S1ð Þ Γ**0**, Γ**<sup>0</sup>** *.*

Later on we succeeded to prove a general result.

**Theorem 6** (L. Bukovský [14])**.** *Assume that* Φ *is one of the symbols* Ω *and* Γ*, and* Ψ *is one of the symbols* O*,* Ω*,* Γ*. Then for any couple* h i Φ, Ψ *different from* h i Ω, O *, a topological space X is an* S1ð Þ Φ, Ψ *-space if and only if* USC*p*ð Þ *X* <sup>þ</sup> *satisfies the selection principle S*1ð Þ Φ**0**, Ψ**<sup>0</sup>** *.*

*Similarly for* Sfin *and U*fin*.*

To describe the selection principles of *Cp*ð Þ *X* we need different covers of the topological space *X*. If *φ* denotes one of the symbols *o*, *ω* or *γ*, then a *φ*-cover U is **shrinkable**, if there exists an open *φ*-cover V such that

$$(\forall V \in \mathcal{V})(\exists U\_V \in \mathcal{U})\,\overline{V} \subseteq U\_V. \tag{5}$$

The family f g *UV* : *V* ∈ V ⊆U is a *φ*-cover as well. The family of all open shrinkable *<sup>φ</sup>*-covers of *<sup>X</sup>* will be denoted by <sup>Φ</sup>*sh*ð Þ *<sup>X</sup>* .

*Covers and Properties of Families of Real Functions DOI: http://dx.doi.org/10.5772/intechopen.100555*

Extending the result by L. Bukovský and J. Haleš [11] for S1 Γ*sh*, Γ we obtain.

**Theorem 7** (L. Bukovský [14])**.** *Assume that* Φ *is one of the symbols* Ω*,* Γ *and* Ψ *is one of the symbols* O*,* Ω*,* Γ*. Then for any couple* h i Φ, Ψ *different from* h i Ω, O *a normal topological space X is an* S1 <sup>Φ</sup>*sh*, <sup>Ψ</sup> *-space if and only if* <sup>C</sup>*p*ð Þ *<sup>X</sup> satisfies the selection principle S*1ð Þ Φ**0**, Ψ**<sup>0</sup>** *.*

*Similarly for* Sfin *and U*fin*.*

The next result is rather a folklore.

**Theorem 8.** *If X is a regular topological space, then for* Φ ¼ O, Ω *and* Ψ ¼ O, Ω, Γ *we have* <sup>Φ</sup>*sh* <sup>¼</sup> <sup>Φ</sup> *and therefore*

$$\mathbf{S}\_1(\boldsymbol{\Phi}^{\rm sh}, \boldsymbol{\Psi}) \equiv \mathbf{S}\_1(\boldsymbol{\Phi}, \boldsymbol{\Psi}).\tag{6}$$

**Corollary 9.** *Let X be a normal topological space. Then for* Φ ¼ O, Ω *and* Ψ ¼ O, Ω, Γ *the following are equivalent:*

a. C*p*ð Þ *X satisfies the selection principle* S1ð Þ Φ**0**, Ψ**<sup>0</sup>**

b. *The family* USCð Þ *X* <sup>þ</sup> *satisfies the sequence selection principle* S1ð Þ Φ**0**, Ψ**<sup>0</sup>** *.*

c. *The family* Cð Þ *X satisfies the sequence selection principle* S1ð Þ Φ**0**, Ψ**<sup>0</sup>** *.*

Note that Theorem 4 is a special case of the Corollary.

#### **4. The measurable covers and functions**

Let <sup>E</sup> <sup>⊆</sup>Pð Þ *<sup>X</sup>* be as above, i.e. <sup>∅</sup>,*<sup>X</sup>* <sup>∈</sup>E. A real function *<sup>f</sup>* <sup>∈</sup>*<sup>X</sup>* is *upper* E*-semimeasurable* if for every real *a*∈ , the set f g *x*∈ *X* : *f x*ð Þ<*a* belongs to E. A real function *<sup>f</sup>* <sup>∈</sup>*<sup>X</sup>* is <sup>E</sup>*-measurable* if for every reals *<sup>a</sup>*<sup>&</sup>lt; *<sup>b</sup>*<sup>∈</sup> , including *<sup>a</sup>* ¼ �∞, *<sup>b</sup>* <sup>¼</sup> <sup>∞</sup>, the set f g *x*∈*X* : *a*<*f x*ð Þ< *b* belongs to E. We denote by USMð Þ *X*, E the set of all real upper E-semimeasurable functions defined on *X*. Similarly, we denote by Mð Þ *X*, E the set of all real E-measurable functions defined on *X*. Note that if E is a *σ*-algebra, then Mð Þ¼ *X*, E USMð Þ *X*, E .

If h i *X*, *τ* is a topological space then Mð Þ¼ *X*, *τ* Cð Þ *X* and USMð Þ¼ *X*, *τ* USCð Þ *X* .

**Theorem 10** (L. Bukovský [15])**.** *Assume that* Φ *is one of the symbols* Ω *or* Γ*,* Ψ *is one of the symbols* O*,* Ω *or* Γ*, and* h i Φ, Ψ 6¼ h i Ω, O *. Let* E *be a family of subsets of a set X,* ∅,*X* ∈E*. If* Ψ ¼ Γ*, we assume that* E *is closed under finite intersections.*


*If* E *is a σ-algebra, then the family* USMð Þ *X*, E <sup>þ</sup> *may be replaced by* Mð Þ *X*, E . For E ¼ *τ* you obtain Theorem 6. For E ¼ BOREL you obtain some results of M. Scheepers and B. Tsaban [16].

#### **5. Countable covers**

A countable family may be considered as a sequence. If U ¼ h i *Un* : *n*∈*ω* is an (open) cover, then <sup>V</sup> <sup>¼</sup> <sup>⋃</sup>*<sup>i</sup>*≤*<sup>n</sup>Ui* : *<sup>n</sup>* <sup>∈</sup>*<sup>ω</sup>* is an (open) *<sup>γ</sup>*-cover. Some covering properties true for U remain true also for V. E.g., sometimes choosing finite subsets of V is same as choosing finite subsets of U.

If U is uncountable you cannot construct the *γ*-cover V. Everything you can do is to construct an (open) *ω*-cover V ¼ f g ⋃W : W ⊆U finite . That is the essence of different behavior of countable and uncountable covers.

W. Just, A. Miller, M. Scheepers and P. Sczeptycki [3] systematically studied the countable covering property G Φctbl ð Þ , Ψ for G ¼ S1, Sfin, *U*fin, Φ, Ψ ¼ O,Λ, Ω, Γ. They obtained several equivalences and as the result the Scheepers Diagram for Countable Covers (**Figure 1**).

Every countable covering property Gð Þ Φ, Ψ , where Φ, Ψ are one of the symbols O,Λ, Ω, Γ and G is one of the symbols S1, Sfin, *U*fin, is equivalent to some covering property in the Scheepers Diagram for Countable Covers. We do not know whether the thick arrow Sfinð Þ! Γ, Ω *U*finð Þ Γ, Ω is reversible. All other arrows of the Diagram are at least consistently not reversible.

#### **6. Sheepers' diagram for arbitrary covers**

The Sheepers' Diagram for Countable Covers is valid only for countable covers. If we allow also uncountable covers, some equivalences, used for simplifying the diagram for countable covers, are generally false. We must make the corresponding corrections.

The first simplification consisted in equivalencies of *U*finð Þ Φ, Ψ for different Φ. Not all of those equivalences are true for uncountable covers. Considering the countable covers, the following properties are equivalent for Ψ ¼ O,Λ, Ω, Γ:

$$U\_{\rm fin}(\mathcal{O}, \Psi) \equiv U\_{\rm fin}(\mathfrak{Q}, \Psi) \equiv U\_{\rm fin}(\Gamma, \Psi). \tag{7}$$

However, allowing uncountable covers we have only

$$U\_{\text{fin}}(\Omega, \Psi) \equiv U\_{\text{fin}}(\mathcal{O}, \Psi) \tag{8}$$

for Ψ ¼ O,Λ, Ω, Γ – see Example 11.

**Figure 1.** *Sheepers' diagram for countable covers.*

*Covers and Properties of Families of Real Functions DOI: http://dx.doi.org/10.5772/intechopen.100555*

In [3] the authors have shown that

$$\mathfrak{S}\_{\mathbf{I}}(\Gamma,\Gamma) \equiv \mathfrak{S}\_{\mathrm{fin}}(\Gamma,\Gamma). \tag{9}$$

The equivalence

$$\mathbf{S}\_1(\mathbf{Q}, \Gamma) \equiv \mathbf{S}\_{\text{fin}}(\mathbf{Q}, \Gamma) \tag{10}$$

remains true also for uncountable covers (an Sfinð Þ Ω, Γ -space is a *γ*-space). One can easily see that for Φ ¼ O, Ω, Γ we have

$$\mathbb{S}\_{\text{fin}}(\Omega,\mathcal{O}) \equiv \mathbb{S}\_{\text{fin}}(\mathcal{O},\mathcal{O}).\tag{11}$$

However, (11) is false at least consistently for Sfinð Þ Γ, Φ and *U*finð Þ Γ, Φ .

**Example 11.** *Assume that* Φ ¼ O, Ω, Γ*. Assume that* b> *ℵ*1*. Then by Theorems 4.6, 4.7 of* [3] *and* (9)*, the discrete space of cardinality ℵ*<sup>1</sup> *is* Sfinð Þ Γ, Φ *and is not Lindelöf. On the other side,* Sfinð Þ O, Φ *and U*finð Þ O, Φ *imply that X is Lindelöf.*

**Example 12.** *No topological space is* Sfinð Þ O,Λ *. So neither* Sfinð Þ O, Ω *nor* Sfinð Þ O, Γ *. Indeed, if X is an infinite Hausdorff topological space, fix a point a*∈*X and an open neighborhood V* 6¼ *X of a. For every x* ∉ *V take an open neighborhood Ux of x not*

**Figure 2.** *Sheepers' Diagram for Arbitrary Infinite Covers.*

*containing a. Then* U ¼ f g *Ux* : *x*∈*X*n*V* ∪f g *V is an open cover of X and no subcover of* U *is a λ-cover.*

Some covering properties are omitted, since they are equivalent with some others included in the **Figure 2**. We present those equivalences. Always the former member of an equivalence is included in the Diagram and the latter member is omitted.

First, take into accunt the equivalences (8)–(11). M. Scheepers [2] in Corollaries 5 and 6 has shown that Sfinð Þ� Γ,Λ *U*finð Þ Γ, O and S1ð Þ� Γ,Λ S1ð Þ Γ, O . Evidently ½*U*finð Þ� Ω,Λ *U*finð Þ� Λ,Λ *U*finð Þ O,Λ .

Taking in account all mentioned results, we obtain the diagram for arbitrary covers (**Figure 2**).

We do not know whether the eleven thick arrows of the Scheepers' Diagram (reversible for countable covers) are reversible for arbitrary covers. The other arrows are not reversible either by Figure 3 of [3] and corresponding results or by Example 11.

#### **7.** *Fσ***-subsets and covering properties**

If U is a family of subsets of *X*, *A* ⊆*X*, we set

$$\mathcal{U}|A = \{U \cap A : U \in \mathcal{U}\}.\tag{12}$$

Let *A* ⊆*X*. If *U* ⊆ *A* is open in the subspace topology, then there exists an open set *<sup>U</sup>* <sup>∗</sup> <sup>⊆</sup>*<sup>X</sup>* such that *<sup>U</sup>* <sup>¼</sup> *<sup>A</sup>* <sup>∩</sup> *<sup>U</sup>* <sup>∗</sup> . Using the Axiom of Choice we choose one such set *<sup>U</sup>* <sup>∗</sup> for each set *U* ⊆ *A* open in *A*. If U is an open (in the subspace topology) cover of *A* then we set <sup>U</sup> <sup>∗</sup> <sup>¼</sup> *<sup>U</sup>* <sup>∗</sup> f g : *<sup>U</sup>* <sup>∈</sup> <sup>U</sup> . Then <sup>U</sup> <sup>¼</sup> <sup>U</sup> <sup>∗</sup> <sup>∣</sup>*A*. Note that <sup>U</sup> <sup>∗</sup> need not be a cover of *<sup>X</sup>*.

By Theorem 3.1 of [3] we have.

**Theorem 13** (W. Just, A. Miller, P. Sczeptycki and M. Scheepers [3])**.** *Let* Φ, Ψ ∈ f g O,Λ, Ω, Γ *,* G *being one of* S1, Sfin, *U*fin*. If a topological space X possesses the covering property* Gð Þ Φ, Ψ *, F* ⊆*X is closed, then F with the subspace topology possesses the property* <sup>G</sup>ð Þ <sup>Φ</sup>, <sup>Ψ</sup> *as well. Moreover, if f* : *<sup>X</sup>* ! *onto Y is continuous, then Y possesses the property* Gð Þ Φ, Ψ *as well.*

Let S be a "topological" property of topological spaces, E ⊆Pð Þ *X* , *X* ∈E. We say that S is *hereditary for* E if assuming that *X* has the property S, every *A* ∈E endowed with the subspace topology has the property S as well. If E ¼ Pð Þ *X* then we simply say that S is *hereditary on X*.

By Theorem 13, for any Φ, Ψ ∈f g O,Λ, Ω, Γ , G being one of S1, Sfin, *U*fin, the covering property Gð Þ Φ, Ψ is hereditary for closed subsets.

In [8], M. Scheepers proved that add Sð Þ <sup>1</sup>ð Þ Γ, Γ ≥h. Since h>*ℵ*0, by Theorem 13 we obtain.

**Corollary 14.** *The covering property* S1ð Þ Γ, Γ *is hereditary for Fσ-subsets of X.* We prove the following result.

**Theorem 15.** *Let* Φ ∈f g O, Ω, Γ *and* Ψ ∈ f g O,Λ, Ω *. Let* G *be one of* S1, Sfin, *U*fin*. The covering property* Gð Þ Φ, Ψ *is hereditary for Fσ-subsets of X.*

**Proof:** Let *F* ¼ ⋃*nFn*, where for each *n* the set *Fn* is closed and *Fn* ⊆*Fn*þ1. Let the countable sequence of open *φ*-covers of *F* be bijectively enumerated as

$$\langle \mathcal{U}\_{n,m} : n, m \in a \rangle. \tag{13}$$

Then U*<sup>n</sup>*,*<sup>m</sup>*∣*Fn* is a *φ*-cover of *Fn*. Apply S1ð Þ Φ, Ψ to the sequence h i U*<sup>n</sup>*,*<sup>m</sup>*j*Fn* : *m* ∈*ω* for every *n*. We obtain sequences h i *Vn*,*<sup>m</sup>* ∈U*<sup>n</sup>*,*<sup>m</sup>*j*Fn* : *m* ∈*ω* such that every family f g *Vn*,*<sup>m</sup>* : *m* ∈*ω* is a *ψ*-cover of *Fn*. Let *Un*,*<sup>m</sup>* ∈U*<sup>n</sup>*,*<sup>m</sup>* be such that *Vn*,*<sup>m</sup>* ¼ *Un*,*<sup>m</sup>* ∩ *Fn*.

One can easily see that the family f g *Un*,*<sup>m</sup>* : *n*, *m* ∈*ω* is a *ψ*-cover of *F*. Since *Un*,*<sup>m</sup>* ∈ U*<sup>n</sup>*,*<sup>m</sup>* for every *n* and *m*, we obtain that *F* possesses the covering property S1ð Þ Φ, Ψ .

For G <sup>¼</sup> Sfin, *<sup>U</sup>*fin the proof is similar. □

#### **8.** *σ***-space**

A topological space *X* is said to be a *σ*-*space*, if every *Gδ*-subset of *X* is an *Fσ*-set. Consequently, every Borel subset of a *σ*-space is an *Fσ*-set.

J. Haleš [4] proved.

**Theorem 16.** *Let X be a perfectly normal topological* S1ð Þ Γ, Γ *-space. The covering property* S1ð Þ Γ, Γ *is hereditary on X if and only if X is a σ-space.*

We obtain an easy Corollary.

**Corollary 17.** *Let* Φ *be one of the symbols* O,Λ, Ω, Γ*. Let* G *be one of* S1, Sfin*. Assume that X is a perfectly normal topological space. If X possesses the covering property* Gð Þ Φ, Γ *and* Gð Þ Φ, Γ *is hereditary on X, then X is a σ-space.*

**Proof:** Note that

$$\mathbf{G}(\Phi,\Gamma) \to \mathbf{G}(\Gamma,\Gamma) \to \mathbf{S}\_1(\Gamma,\Gamma),\tag{14}$$

(if G <sup>¼</sup> Sfin use (9)) and <sup>Γ</sup><sup>⊆</sup> <sup>Φ</sup>. □ Following Haleš' proof of Theorem 16 we obtain.

**Theorem 18.** *Let X be a σ-space,* Φ *being one of the symbols* O,Λ*ctbl*, Γ *and* Ψ *being one of* O,Λ, Ω, Γ*. Let* G *be one of* S1, Sfin, *U*fin*. Then* Gð Þ Φ, Ψ *is hereditary on X.*

**Proof:** Assume that *X* is a topological Gð Þ Φ, Ψ -space, where Φ ¼ O,Λ*ctbl*, Γ and Ψ ¼ O,Λ, Ω, Γ. Assume also that *X* is a *σ*-space and *A* ⊆ *X*. Let h i U*<sup>n</sup>* : *n* ∈*ω* be a sequence of open (in the subspace topology) *φ*-covers of *A*.

If <sup>Φ</sup> <sup>¼</sup> <sup>O</sup> then <sup>U</sup> <sup>∗</sup> *<sup>n</sup>* <sup>∣</sup>*<sup>B</sup>* is an *<sup>o</sup>*-cover of *<sup>B</sup>* <sup>¼</sup> <sup>⋂</sup>*n*⋃<sup>U</sup> <sup>∗</sup> *<sup>n</sup>* ⊇ *A*. Since *B* is a Borel set, therefore an *Fσ*-set in *X*, by Theorem 15, the set *B* possesses the covering property <sup>G</sup>ð Þ <sup>O</sup>, <sup>Ψ</sup> . So, for each *<sup>n</sup>* <sup>∈</sup>*<sup>ω</sup>* there exists a *Un* <sup>∈</sup><sup>U</sup> <sup>∗</sup> *<sup>n</sup>* ∣*B* or finite set W*<sup>n</sup>* ⊆ U <sup>∗</sup> *<sup>n</sup>* ∣*B* such that f g *Un* : *n*∈*ω* , or ⋃*n*W*n*, or f g ⋃W*<sup>n</sup>* : *n* ∈*ω* is a *ψ*-cover of *B*, respectively. Then f g *Un* : *n*∈*ω* ∣*A*, or ⋃*n*W*n*∣*A*, or f g ⋃W*<sup>n</sup>* : *n*∈*ω* ∣*A* is an open *ψ*-cover of *A*, respectively.

Let Φ ¼ Λ*ctbl*. Since each U*<sup>n</sup>* is countable, we can assume that f g *Un*,*<sup>m</sup>* : *m* ∈*ω* is a bijective enumeration of U*n*. The family U*<sup>n</sup>*,*<sup>m</sup>* ¼ f g *Un*,*<sup>k</sup>* : *k*≥ *m* is a *λ*-cover of *A*. If we set *<sup>B</sup>* <sup>¼</sup> <sup>⋂</sup>*<sup>n</sup>*,*<sup>m</sup>*⋃<sup>U</sup> <sup>∗</sup> *<sup>n</sup>*,*<sup>m</sup>* ⊇ *A* then each U <sup>∗</sup> *<sup>n</sup>* ∣*B* is a *λ*-cover of *B*. Continue as above.

If Φ ¼ Γ we can assume that each U*<sup>n</sup>* is countable. For every *n*∈*ω* let f g *Un*,*<sup>m</sup>* : *<sup>m</sup>* <sup>∈</sup> *<sup>ω</sup>* be a bijective enumeration of <sup>U</sup>*n*. Then <sup>U</sup> <sup>∗</sup> *<sup>n</sup>* ∣*B* is a *γ*-cover of *B* ¼ ⋂*n*⋃*<sup>k</sup>* ∩ *<sup>m</sup>* <sup>≥</sup>*kU* <sup>∗</sup> *<sup>n</sup>*,*<sup>m</sup>* ⊇ *A*. The set *B* is Borel, therefore *Fσ*. By Theorem 15, *B* is an <sup>G</sup>ð Þ <sup>Γ</sup>, <sup>Ψ</sup> -space. Continue as above. □

For Φ ¼ Ψ ¼ Γ, G ¼ S1, we obtain one implication of the Haleš' Theorem 16.

**Corollary 19.** *Let* Φ *be one of the symbols* O,Λ*ctbl*, Γ*. Let* G *be one of* S1, Sfin*. The covering property* Gð Þ Φ, Γ *is hereditary on a perfectly normal topological space X if and only if X is a σ-space.*

**Corollary 20.** *Let* Φ *be one of the symbols* O,Λ*ctbl*, Γ*. Let* G *be one of* S1, Sfin*. Assume that a perfectly normal topological space X possesses the covering property* Gð Þ Φ, Γ *. Then X is hereditary* Gð Þ Φ, Γ *if and only if X is hereditary* S1ð Þ Γ, Γ *.*

#### **9. Remarks**

I have obtained a short time for writing this paper. So, I have no time to collect all results known before proving Theorems 5, 6, 7, 10, 15 and 18. For a partial presentation of such known results see [14, 15].

*Advanced Topics of Topology*

### **Classification**

*2010 MSC:* 54C35, 54C20, 54D55

### **Author details**

Lev Bukovský† Institute of Mathematics, P.J. Šafárik University, Košice, Slovakia

\*Address all correspondence to: lev.bukovsky@upjs.sk

† Deceased.

© 2022 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Covers and Properties of Families of Real Functions DOI: http://dx.doi.org/10.5772/intechopen.100555*

#### **References**

[1] Bukovský L., *On* wQN\* *and* wQN\* *spaces,* Topology Appl. **156** (2008), 24–27.

[2] Scheepers M., *Combinatorics of open covers* I*: Ramsey theory,* Topology Appl. **69** (1996), 31–62.

[3] Just W., Miller A.W., Scheepers M. and Szeptycki P.J., *Combinatorics of open covers II,* Topology Appl. **73** (1996), 241–266.

[4] Haleš J., *On Scheepers' Conjecture,* Acta Univ. Carolinae – Math. Phys. **46** (2005), 27–31.

[5] Engelking R., General Topology, Monografie Matematyczne **60**, Warszawa 1977, revised edition Heldermann Verlag, Berlin 1989.

[6] Blass A. Combinatorial cardinal characteristics of the continuum. In: Foreman M, Kanamori A, editors. Handbook of Set Theory. Dordrecht: Springer; 2010. pp. 24-27

[7] van Douwen E. The integers and topology. In: Kunen K, Vaughan J, editors. Handbook of Set-Theoretic Topology. North-Holland; 1984. pp. 111-167

[8] Scheepers M., *A sequential property of* C*p(X) and a covering property of Hurewicz,* Proc. Amer. Math. Soc. **125** (1997), 2789–2795.

[9] Hurewicz W., *Über Folgen stetiger Funktionen,* Fund. Math. **9** (1927), 193–204.

[10] Menger K., *Einige Überdeckungssatze der Punktmengenlehre,* Sitzungsberichte. Abt. 2a, Mathematik, Astronomie, Physik, Meteorologie und Mechanik (Wiener Akademie) **133** (1924), 421–444.

[11] Bukovský L. and Haleš J., *On Hurewicz Properties,* Topology Appl. **132** (2003), 71-79.

[12] Gerlits F. and Nagy Z., *Some properties of* C*(X), part I,* Topology Appl. **14** (1982), 151–161.

[13] Sakai M., *Property* C" *and function spaces,* Proc. Amer. Math. Soc. **104** (1988), 917–919.

[14] Bukovský L., *Selection principle* S1 *and combinatorics of open covers,* Topology Appl. **258** (2019), 239–252.

[15] Bukovský L., *Measurable Functions and Covering Properties,* Topology Appl. **258** (2021), 107787.

[16] Scheepers M. and Tsaban B.,*The combinatorics of Borel covers,* Topology Appl. **121** (2002), 357–382.

Section 4
