Covers and Properties of Families of Real Functions

Abstract

We present results on the relationships of the covering property GΦΨ for Φ,Ψ∈OΛΩΓ and G∈S1SfinUfin of a topological space and the selection property GΦ0Ψ0 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.

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 $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ of a topological space $X$ and the selection property $\mathrm{G}\left({\mathrm{\Phi }}_{\mathbf{0}},{\mathrm{\Psi }}_{\mathbf{0}}\right)$ of the topological space of upper semicontinuous real functions ${\mathrm{USC}}_p\left(X\right)$ on $X$, for $\mathrm{\Phi },\mathrm{\Psi }\in \left\{\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }\right\}$ and $\mathrm{G}\in \left\{{\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}},{\mathit{U}}_{\mathrm{fin}}\right\}$. The upper semicontinuous functions in this connection were for first time used in [1]. In some important cases we can replace the space ${\mathrm{USC}}_p\left(X\right)$ by the topological space ${\mathrm{C}}_p\left(X\right)$ of continuous functions. So we obtain the equivalence of some topological property of ${\mathrm{C}}_p\left(X\right)$ 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 $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ 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 $F_{\sigma }$-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 $\sigma$-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 $⟨X,\tau ⟩$, where $\tau$ is the topology on $X$: the set of all open subsets of $X$ [5]. The smallest $\sigma$-algebra containing all open sets is the family $\text{BOREL}\left(X\right)=\text{BOREL}$ of Borel subsets of $X$.

We recall some covering properties which were introduced by M. Scheepers in [2]. If $\mathcal{A},\mathcal{B}\subseteq \mathcal{P}\left(Y\right)$ are sets of subsets of a set $Y$, then the covering principle ${\mathrm{S}}_1\left(\mathcal{A},\mathcal{B}\right)$ means the following: for every sequence $⟨{\mathcal{U}}_n:n\in \omega ⟩$ of elements of $\mathcal{A}$ and for every $n\in \omega$ there exists a $U_n\in {\mathcal{U}}_n$ such that $\left\{U_n:n\in \omega \right\}\in \mathcal{B}$. The covering principles ${\mathrm{S}}_{\mathrm{fin}}\left(\mathcal{A},\mathcal{B}\right)$ and ${\mathit{U}}_{\mathrm{fin}}\left(\mathcal{A},\mathcal{B}\right)$ are there defined in a similar way. In the case of ${\mathrm{S}}_{\mathrm{fin}}\left(\mathcal{A},\mathcal{B}\right)$ we choose finite sets ${\mathcal{W}}_n\subseteq {\mathcal{U}}_n$ such that ${\bigcup }_n{\mathcal{W}}_n\in \mathcal{B}$. In the case of ${\mathrm{S}}_{\mathrm{fin}}\left(\mathcal{A},\mathcal{B}\right)$ we chose finite sets ${\mathcal{W}}_n\subseteq {\mathcal{U}}_n$ such that ${\bigcup }_n{\mathcal{W}}_n\in \mathcal{B}$. In the case of ${\mathit{U}}_{\mathrm{fin}}\left(\mathcal{A},\mathcal{B}\right)$ we chose finite sets ${\mathcal{W}}_n\subseteq {\mathcal{U}}_n$ such that $\left\{\bigcup {\mathcal{W}}_n:n\in \omega \right\}\in \mathcal{B}$. Actually we tacitly assume that $Y=\mathcal{P}\left(X\right)$. If $Y{\subseteq }^X\mathbb{R}$ we define the selection property ${\mathit{U}}_{\mathrm{fin}}^{\ast }\left(\mathcal{A},\mathcal{B}\right)$ similarly, but we ask¹ that $\left\{min{\mathcal{W}}_n:n\in \omega \right\}\in \mathcal{B}$.

A family $\mathcal{U}$ of subsets of $X$ is a cover of $X$ if $\bigcup \mathcal{U}=X$. A cover is open, if every element of the cover is an open set. A cover $\mathcal{V}\subseteq \mathcal{U}$ is said to be a subcover of the cover $\mathcal{U}$. If we deal with a countable cover of $X$ we can consider it as a sequence of subsets.

If $\mathcal{A}$ is a family of subsets of $X$, then we denote by ${\mathcal{A}}_{\text{ctbl}}$ the family of all countable elements of $\mathcal{A}$. If $\mathcal{A}{\subseteq }^X\mathbb{R}$ then ${\mathcal{A}}^{+}$ is the family $\left\{f\in \mathcal{A}:\left(\forall x\in X\right)f\left(x\right)\ge 0\right\}$.

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 $\mathcal{U}$ of a set $X$ is a $\lambda$-cover if for every $x\in X$ the set $\left\{U\in \mathcal{U}:x\in U\right\}$ is infinite. A cover $\mathcal{U}$ of a set $X$ is an $\omega$-cover if $X\notin \mathcal{U}$ and for every finite $F\subseteq X$ there exists a $U\in \mathcal{U}$ such that $F\subseteq U$. Finally, a cover $\mathcal{U}$ of a set $X$ is a $\gamma$-cover if $\mathcal{U}$ is infinite and for every $x\in X$ the set $\left\{U\in \mathcal{U}:x\notin U\right\}$ is finite.

We shall use the following convention. If the lower case letters $\phi$ or $\psi$ denote one of the symbols $o$, $\lambda$, $\omega$ or $\gamma$, then the capital letters $\mathrm{\Phi }$ or $\mathrm{\Psi }$ denote the corresponding symbol $\mathcal{O}$, $\mathrm{\Lambda }$, $\mathrm{\Omega }$ or $\mathrm{\Gamma }$, respectively, and vice versa.

Assume that $\mathcal{E}\subseteq \mathcal{P}\left(X\right)$ and $\varnothing ,X\in \mathcal{E}$. Dealing with the covering property ${\mathit{U}}_{\mathrm{fin}}$ we assume that $\mathcal{E}$ is closed under finite unions.

Let $\phi \in \left\{o,\lambda ,\omega ,\gamma \right\}$. We denote by $\mathrm{\Phi }\left(\mathcal{E}\right)$ the family of all $\phi$-covers $\mathcal{U}$ of $X$ satisfying $\mathcal{U}\subseteq \mathcal{E}$. If $⟨X,\tau ⟩$ is an infinite Hausdorff topological space, then $\mathrm{\Phi }\left(\tau \right)$ is simply denoted as $\mathrm{\Phi }$.

Evidently (we should eventually omit $X$ from a $\gamma$-cover)

$$\mathrm{\Gamma }\left(\mathcal{E}\right)\subseteq \mathrm{\Omega }\left(\mathcal{E}\right)\subseteq \mathrm{\Lambda }\left(\mathcal{E}\right)\subseteq \mathcal{O}\left(\mathcal{E}\right).$$E1

We say that a family $\mathcal{V}\subseteq \mathcal{P}\left(X\right)$ is a refinement of the family $\mathcal{U}\subseteq \mathcal{P}\left(X\right)$ if

$$\left(\forall V\in \mathcal{V}\right)\left(\exists U\in \mathcal{U}\right)V\subseteq U.$$E2

Let the family $\mathcal{V}\subseteq \mathcal{P}\left(X\right)$ be a refinement of the family $\mathcal{U}\subseteq \mathcal{P}\left(X\right)$. If $\mathcal{V}$ is an $o$- or an $\omega$-cover, then $\mathcal{U}$ is such a cover as well. This is not true for $\lambda$- and $\gamma$-covers. If we add finitely many subsets of $X$ to a $\phi$-cover, $\phi \in \left\{o,\lambda ,\omega ,\gamma \right\}$, we obtain a $\phi$-cover. Any infinite subset of a $\gamma$-cover is a $\gamma$-cover as well.

A set ${}^{\omega }\mathbb{R}$ of all sequences of reals is quasi-ordered by the eventual dominating relation

$$\phi {\le }^{\ast }\psi \equiv \left(\exists n_0\in \omega \right)\left(\forall n\ge n_0\right)\phi \left(n\right)\le \psi \left(n\right).$$E3

A set $\mathcal{F}{\subseteq }^{\omega }\mathbb{R}$ is called bounded if there exists a sequence $\psi {\in }^{\omega }\mathbb{R}$ such that $\phi {\le }^{\ast }\psi$ for every $\phi \in \mathcal{F}$. The set $\mathcal{F}$ is dominating if for every $\psi {\in }^{\omega }\mathbb{R}$ there exists a $\phi \in \mathcal{F}$ such that $\psi {\le }^{\ast }\phi$. The cardinals $\mathfrak{b}$ and $\mathfrak{d}$ (see e.g. [6, 7]) are the smallest cardinalities of an unbounded and dominating family, respectively.

The set ${}^X\mathbb{R}$ is endowed with the product topology. For any real $a\in \mathbb{R}$ we denote by $\mathbf{a}$ the constant function defined on $X$ with value $a$. There are at least two important subfamilies of ${}^X\mathbb{R}$: the family $C\left(X\right)$ of all continuous functions and the family $\mathrm{USC}\left(X\right)$ of all upper semicontinuous functions². If they are endowed with the product topology we write ${\mathrm{C}}_p\left(X\right)$ and ${\mathrm{USC}}_p\left(X\right)$, respectively. Note that $f_n\to f$ in the product topology if and only if $\left(\forall x\in X\right)f_n\left(x\right)\to f\left(x\right)$.

We introduce three properties of an infinite family $F{\subseteq }^X\mathbb{R}$ of real functions.

$$\begin{array}{ll}\left({\mathcal{O}}_{\mathbf{0}}\right)& \mathbf{0}\left(x\right)=0\in \overline{\left\{f\left(x\right):f\in F\right\}}\mathit{\text{for}}\mathit{\text{every}}x\in X.\\ \left({\mathrm{\Omega }}_{\mathbf{0}}\right)& \mathbf{0}\notin F\mathit{\text{and}}\mathbf{0}\in \overline{F}\mathit{\text{in}}\mathit{\text{the}}\mathit{\text{prodiuct}}\mathit{\text{topology}}{\mathit{\text{of}}}^X\mathbb{R}.\\ \left({\mathrm{\Gamma }}_{\mathbf{0}}\right)& F\mathit{\text{is}}\mathit{\text{infinite}}\mathit{\text{and}}\mathit{\text{for}}\mathit{\text{every}}\epsilon >0\mathit{\text{and}}\mathit{\text{for}}\\ & \mathit{\text{every}}x\in X\mathit{\text{the}}\mathit{set}\left\{f\in F:|f\left(x\right)|\ge \epsilon \right\}\mathit{\text{is}}\mathit{\text{finite}}.\end{array}$$

Let $\mathrm{\Phi }$ and $\mathrm{\Psi }$ be one of the symbols $\mathcal{O},\mathrm{\Omega },\mathrm{\Gamma }$. Similarly as for covers, we define for an infinite family $F{\subseteq }^X\mathbb{R}$ of real functions the set

$${\mathrm{\Phi }}_{\mathbf{0}}\left(F\right)=\left\{H\subseteq F:H\text{is}\text{infinite}\text{and}\mathrm{has}\text{the}\text{property}\left({\mathrm{\Phi }}_{\mathbf{0}}\right)\right\}.$$E4

$F$ satisfies the selection principle ${\mathsf{S}}_1\left({\mathrm{\Phi }}_{\mathrm{0}},{\mathrm{\Psi }}_{\mathrm{0}}\right)$ if ${\mathsf{S}}_1\left({\mathrm{\Phi }}_{\mathrm{0}}\left(F\right),{\mathrm{\Psi }}_{\mathrm{0}}\left(F\right)\right)$ holds. Identifying the countable sets of functions with sequences of functions, we say that $F$ satisfies the sequence selection principle ${\mathsf{S}}_1\left({\mathrm{\Phi }}_{\mathrm{0}},{\mathrm{\Psi }}_{\mathrm{0}}\right)$ if ${\mathsf{S}}_1\left({\left({\mathrm{\Phi }}_{\mathrm{0}}\left(F\right)\right)}_{\text{ctbl}},{\left({\mathrm{\Psi }}_{\mathrm{0}}\left(F\right)\right)}_{\text{ctbl}}\right)$ holds (compare, e.g., [8]: for each sequence of sequences $⟨⟨f_{n,m}:m\in \omega ⟩:n\in \omega ⟩$ of functions from $F$ with the property $\left({\mathrm{\Phi }}_{\mathbf{0}}\right)$, there exists a functions $\alpha {\in }^{\omega }\omega$ such that $\left\{f_{n,\alpha \left(n\right)}:n\in \omega \right\}$ has the property $\left({\mathrm{\Psi }}_{\mathbf{0}}\right)$.) Similarly for ${\mathrm{S}}_{\mathrm{fin}}$ and ${\mathit{U}}_{\mathrm{fin}}$.

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.

1. $X$ is a ${\mathit{U}}_{\mathrm{fin}}\left({\mathcal{O}}_{\text{ctbl}},\mathcal{O}\right)$

2. For every sequence $⟨f_n:n\in \omega ⟩$ of continuous real functions the family $\left\{⟨f_n\left(x\right):n\in \omega ⟩:x\in X\right\}{\subseteq }^{\omega }\mathbb{R}$ is bounded.

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

1. $X$ is a ${\mathit{U}}_{\mathrm{fin}}\left({\mathcal{O}}_{\text{ctbl}},\mathrm{\Gamma }\right)$

2. For every sequence $⟨f_n:n\in \omega ⟩$ of continuous real functions the family $\left\{⟨f_n\left(x\right):n\in \omega ⟩:x\in X\right\}{\subseteq }^{\omega }\mathbb{R}$ is dominating.

Note that the property ${\mathit{U}}_{\mathrm{fin}}\left({\mathcal{O}}_{\text{ctbl}},\mathrm{\Gamma }\right)$ 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 $\gamma$-space if every open $\omega$-cover of $X$ has a countable $\gamma$-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:

1. ${\mathrm{C}}_p\left(X\right)$ is Fréchet.

2. $X$ is a $\gamma$-space.

3. $X$ is an ${\mathrm{S}}_1\left(\mathrm{\Omega },\mathrm{\Gamma }\right)$-space.

A topological space $X$ has countable strong fan tightness if $A_n\subseteq X$ and $x\in \overline{A_n}$, $n\in \omega$ imply that there exists a sequence $⟨x_n:n\in \omega ⟩$ such that $x_n\in A_n$ and $x\in \overline{\left\{x_n:n\in \omega \right\}}$.

Theorem 4 (M. Sakai [13]). A Tychonoff topological space has the covering property ${\mathrm{S}}_1\left(\mathrm{\Omega },\mathrm{\Omega }\right)$ if and only if the topological space ${\mathrm{C}}_p\left(X\right)$ has countable strong fan tightness.

That was M. Scheepers [2] who began the systematic study of the covering properties $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ for $\mathrm{G}={\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}},{\mathit{U}}_{\mathrm{fin}}$, $\mathrm{\Phi },\mathrm{\Psi }=\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }$.

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 ${\mathrm{S}}_1\left(\mathrm{\Gamma },\mathrm{\Gamma }\right)$-space if and only if ${\mathrm{USC}}_p{\left(X\right)}^{+}$ satisfies the selection principle ${\mathrm{S}}_1\left({\mathrm{\Gamma }}_{\mathbf{0}},{\mathrm{\Gamma }}_{\mathbf{0}}\right)$.

Later on we succeeded to prove a general result.

Theorem 6 (L. Bukovský [14]). Assume that $\mathrm{\Phi }$ is one of the symbols $\mathrm{\Omega }$ and $\mathrm{\Gamma }$, and $\mathrm{\Psi }$ is one of the symbols $\mathcal{O}$, $\mathrm{\Omega }$, $\mathrm{\Gamma }$. Then for any couple $⟨\mathrm{\Phi },\mathrm{\Psi }⟩$ different from $⟨\mathrm{\Omega },\mathcal{O}⟩$, a topological space $X$ is an ${\mathrm{S}}_1\left(\mathrm{\Phi },\mathrm{\Psi }\right)$-space if and only if ${\mathrm{USC}}_p{\left(X\right)}^{+}$ satisfies the selection principle $S_1\left({\mathrm{\Phi }}_{\mathbf{0}},{\mathrm{\Psi }}_{\mathbf{0}}\right)$.

Similarly for ${\mathrm{S}}_{\mathrm{fin}}$ and ${\mathit{U}}_{\mathrm{fin}}$.

To describe the selection principles of $C_p\left(X\right)$ we need different covers of the topological space $X$. If $\phi$ denotes one of the symbols $o$, $\omega$ or $\gamma$, then a $\phi$-cover $\mathcal{U}$ is shrinkable, if there exists an open $\phi$-cover $\mathcal{V}$ such that

$$\left(\forall V\in \mathcal{V}\right)\left(\exists U_V\in \mathcal{U}\right)\overline{V}\subseteq U_V.$$E5

The family $\left\{U_V:V\in \mathcal{V}\right\}\subseteq \mathcal{U}$ is a $\phi$-cover as well. The family of all open shrinkable $\phi$-covers of $X$ will be denoted by ${\mathrm{\Phi }}^{\mathit{sh}}\left(X\right)$.

Extending the result by L. Bukovský and J. Haleš [11] for ${\mathrm{S}}_1\left({\mathrm{\Gamma }}^{\mathit{sh}},\mathrm{\Gamma }\right)$ we obtain.

Theorem 7 (L. Bukovský [14]). Assume that $\mathrm{\Phi }$ is one of the symbols $\mathrm{\Omega }$, $\mathrm{\Gamma }$ and $\mathrm{\Psi }$ is one of the symbols $\mathcal{O}$, $\mathrm{\Omega }$, $\mathrm{\Gamma }$. Then for any couple $⟨\mathrm{\Phi },\mathrm{\Psi }⟩$ different from $⟨\mathrm{\Omega },\mathcal{O}⟩$ a normal topological space $X$ is an ${\mathrm{S}}_1\left({\mathrm{\Phi }}^{\mathit{sh}},\mathrm{\Psi }\right)$-space if and only if ${\mathrm{C}}_p\left(X\right)$ satisfies the selection principle $S_1\left({\mathrm{\Phi }}_{\mathbf{0}},{\mathrm{\Psi }}_{\mathbf{0}}\right)$.

Similarly for ${\mathrm{S}}_{\mathrm{fin}}$ and ${\mathit{U}}_{\mathrm{fin}}$.

The next result is rather a folklore.

Theorem 8. If $X$ is a regular topological space, then for $\mathrm{\Phi }=\mathcal{O},\mathrm{\Omega }$ and $\mathrm{\Psi }=\mathcal{O},\mathrm{\Omega },\mathrm{\Gamma }$ we have ${\mathrm{\Phi }}^{\mathit{sh}}=\mathrm{\Phi }$ and therefore

$${\mathrm{S}}_1\left({\mathrm{\Phi }}^{\mathit{sh}},\mathrm{\Psi }\right)\equiv {\mathrm{S}}_1\left(\mathrm{\Phi },\mathrm{\Psi }\right).$$E6

Corollary 9. Let $X$ be a normal topological space. Then for $\mathrm{\Phi }=\mathcal{O},\mathrm{\Omega }$ and $\mathrm{\Psi }=\mathcal{O},\mathrm{\Omega },\mathrm{\Gamma }$ the following are equivalent:

1. ${\mathrm{C}}_p\left(X\right)$ satisfies the selection principle ${\mathrm{S}}_1\left({\mathrm{\Phi }}_{\mathbf{0}},{\mathrm{\Psi }}_{\mathbf{0}}\right)$

2. The family $\mathrm{USC}{\left(X\right)}^{+}$ satisfies the sequence selection principle ${\mathrm{S}}_1\left({\mathrm{\Phi }}_{\mathbf{0}},{\mathrm{\Psi }}_{\mathbf{0}}\right)$.

3. The family $\mathrm{C}\left(X\right)$ satisfies the sequence selection principle ${\mathrm{S}}_1\left({\mathrm{\Phi }}_{\mathbf{0}},{\mathrm{\Psi }}_{\mathbf{0}}\right)$.

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

4. The measurable covers and functions

Let $\mathcal{E}\subseteq \mathcal{P}\left(X\right)$ be as above, i.e. $\varnothing ,X\in \mathcal{E}$. A real function $f{\in }^X\mathbb{R}$ is upper $\mathcal{E}$-semimeasurable if for every real $a\in \mathbb{R}$, the set $\left\{x\in X:f\left(x\right)<a\right\}$ belongs to $\mathcal{E}$. A real function $f{\in }^X\mathbb{R}$ is $\mathcal{E}$-measurable if for every reals $a<b\in \mathbb{R}$, including $a=-\infty ,b=\infty$, the set $\left\{x\in X:a<f\left(x\right)<b\right\}$ belongs to $\mathcal{E}$. We denote by $\mathrm{USM}\left(X,\mathcal{E}\right)$ the set of all real upper $\mathcal{E}$-semimeasurable functions defined on $X$. Similarly, we denote by $\mathrm{M}\left(X,\mathcal{E}\right)$ the set of all real $\mathcal{E}$-measurable functions defined on $X$. Note that if $\mathcal{E}$ is a $\sigma$-algebra, then $\mathrm{M}\left(X,\mathcal{E}\right)=\mathrm{USM}\left(X,\mathcal{E}\right)$.

If $⟨X,\tau ⟩$ is a topological space then $\mathrm{M}\left(X,\tau \right)=\mathrm{C}\left(X\right)$ and $\mathrm{USM}\left(X,\tau \right)=\mathrm{USC}\left(X\right)$.

Theorem 10 (L. Bukovský [15]). Assume that $\mathrm{\Phi }$ is one of the symbols $\mathrm{\Omega }$ or $\mathrm{\Gamma }$, $\mathrm{\Psi }$ is one of the symbols $\mathcal{O}$, $\mathrm{\Omega }$ or $\mathrm{\Gamma }$, and $⟨\mathrm{\Phi },\mathrm{\Psi }⟩\ne ⟨\mathrm{\Omega },\mathcal{O}⟩$. Let $\mathcal{E}$ be a family of subsets of a set $X$, $\varnothing ,X\in \mathcal{E}$. If $\mathrm{\Psi }=\mathrm{\Gamma }$, we assume that $\mathcal{E}$ is closed under finite intersections.

1. $X$ possesses the covering property ${\mathrm{S}}_1\left(\mathrm{\Phi }\left(\mathcal{E}\right),\mathrm{\Psi }\left(\mathcal{E}\right)\right)$ if and only if $\mathrm{USM}{\left(X,\mathcal{E}\right)}^{+}$ satisfies the selection principle ${\mathrm{S}}_1\left({\mathrm{\Phi }}_{\mathbf{0}},{\mathrm{\Psi }}_{\mathbf{0}}\right)$.

2. Similarly for ${\mathrm{S}}_{\mathrm{fin}}$.

3. If $\mathcal{E}$ is closed under finite unions, then $X$ possesses the covering property ${\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Phi }\left(\mathcal{E}\right),\mathrm{\Psi }\left(\mathcal{E}\right)\right)$ if and only if $\mathrm{USM}{\left(X,\mathcal{E}\right)}^{+}$ satisfies the selection principle ${\mathit{U}}_{\mathrm{fin}}^{\ast }\left({\mathrm{\Phi }}_{\mathbf{0}},{\mathrm{\Psi }}_{\mathbf{0}}\right)$.

If $\mathcal{E}$ is a $\sigma$-algebra, then the family $\mathrm{USM}{\left(X,\mathcal{E}\right)}^{+}$ may be replaced by $\mathrm{M}\left(X,\mathcal{E}\right)$.

For $\mathcal{E}=\tau$ you obtain Theorem 6. For $\mathcal{E}=\text{BOREL}$ you obtain some results of M. Scheepers and B. Tsaban [16].

5. Countable covers

A countable family may be considered as a sequence. If $\mathcal{U}=⟨U_n:n\in \omega ⟩$ is an (open) cover, then $\mathcal{V}=⟨{\bigcup }_{i\le n}{U}_i:n\in \omega ⟩$ is an (open) $\gamma$-cover. Some covering properties true for $\mathcal{U}$ remain true also for $\mathcal{V}$. E.g., sometimes choosing finite subsets of $\mathcal{V}$ is same as choosing finite subsets of $\mathcal{U}$.

If $\mathcal{U}$ is uncountable you cannot construct the $\gamma$-cover $\mathcal{V}$. Everything you can do is to construct an (open) $\omega$-cover $\mathcal{V}=\left\{\bigcup \mathcal{W}:\mathcal{W}\subseteq \mathcal{U}\text{finite}\right\}$. 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 $\mathrm{G}\left({\mathrm{\Phi }}_{\text{ctbl}},\mathrm{\Psi }\right)$ for $\mathrm{G}={\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}},{\mathit{U}}_{\mathrm{fin}}$, $\mathrm{\Phi },\mathrm{\Psi }=\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }$. They obtained several equivalences and as the result the Scheepers Diagram for Countable Covers (Figure 1).

Figure 1.

Sheepers’ diagram for countable covers.

Every countable covering property $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$, where $\mathrm{\Phi },\mathrm{\Psi }$ are one of the symbols $\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }$ and $\mathrm{G}$ is one of the symbols ${\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}},{\mathit{U}}_{\mathrm{fin}}$, is equivalent to some covering property in the Scheepers Diagram for Countable Covers. We do not know whether the thick arrow ${\mathrm{S}}_{\mathrm{fin}}\left(\mathrm{\Gamma },\mathrm{\Omega }\right)\to {\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Gamma },\mathrm{\Omega }\right)$ 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 ${\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ for different $\mathrm{\Phi }$. Not all of those equivalences are true for uncountable covers. Considering the countable covers, the following properties are equivalent for $\mathrm{\Psi }=\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }$:

$${\mathit{U}}_{\mathrm{fin}}\left(\mathcal{O},\mathrm{\Psi }\right)\equiv {\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Omega },\mathrm{\Psi }\right)\equiv {\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Gamma },\mathrm{\Psi }\right).$$E7

However, allowing uncountable covers we have only

$${\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Omega },\mathrm{\Psi }\right)\equiv {\mathit{U}}_{\mathrm{fin}}\left(\mathcal{O},\mathrm{\Psi }\right)$$E8

for $\mathrm{\Psi }=\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }$ – see Example 11.

In [3] the authors have shown that

$${\mathrm{S}}_1\left(\mathrm{\Gamma },\mathrm{\Gamma }\right)\equiv {\mathrm{S}}_{\mathrm{fin}}\left(\mathrm{\Gamma },\mathrm{\Gamma }\right).$$E9

The equivalence

$${\mathrm{S}}_1\left(\mathrm{\Omega },\mathrm{\Gamma }\right)\equiv {\mathrm{S}}_{\mathrm{fin}}\left(\mathrm{\Omega },\mathrm{\Gamma }\right)$$E10

remains true also for uncountable covers (an ${\mathrm{S}}_{\mathrm{fin}}\left(\mathrm{\Omega },\mathrm{\Gamma }\right)$-space is a $\gamma$-space). One can easily see that for $\mathrm{\Phi }=\mathcal{O},\mathrm{\Omega },\mathrm{\Gamma }$ we have

$${\mathrm{S}}_{\mathrm{fin}}\left(\mathrm{\Omega },\mathcal{O}\right)\equiv {\mathrm{S}}_{\mathrm{fin}}\left(\mathcal{O},\mathcal{O}\right).$$E11

However, (11) is false at least consistently for ${\mathrm{S}}_{\mathrm{fin}}\left(\mathrm{\Gamma },\mathrm{\Phi }\right)$ and ${\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Gamma },\mathrm{\Phi }\right)$.

Example 11. Assume that $\mathrm{\Phi }=\mathcal{O},\mathrm{\Omega },\mathrm{\Gamma }$. Assume that $\mathfrak{b}>{\aleph }_1$. Then by Theorems 4.6, 4.7 of [3] and (9), the discrete space of cardinality ${\aleph }_1$ is ${\mathrm{S}}_{\mathrm{fin}}\left(\mathrm{\Gamma },\mathrm{\Phi }\right)$ and is not Lindelöf. On the other side, ${\mathrm{S}}_{\mathrm{fin}}\left(\mathcal{O},\mathrm{\Phi }\right)$ and ${\mathit{U}}_{\mathrm{fin}}\left(\mathcal{O},\mathrm{\Phi }\right)$ imply that $X$ is Lindelöf.

Example 12. No topological space is ${\mathrm{S}}_{\mathrm{fin}}\left(\mathcal{O},\mathrm{\Lambda }\right)$. So neither ${\mathrm{S}}_{\mathrm{fin}}\left(\mathcal{O},\mathrm{\Omega }\right)$ nor ${\mathrm{S}}_{\mathrm{fin}}\left(\mathcal{O},\mathrm{\Gamma }\right)$. Indeed, if $X$ is an infinite Hausdorff topological space, fix a point $a\in X$ and an open neighborhood $V\ne X$ of $a$. For every $x\notin V$ take an open neighborhood $U_x$ of $x$ not containing $a$. Then $\mathcal{U}=\left\{U_x:x\in X\backslash V\right\}\cup \left\{V\right\}$ is an open cover of $X$ and no subcover of $\mathcal{U}$ is a $\lambda$-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.

Figure 2.

Sheepers’ Diagram for Arbitrary Infinite Covers.

First, take into accunt the equivalences (8)–(11). M. Scheepers [2] in Corollaries 5 and 6 has shown that ${\mathrm{S}}_{\mathrm{fin}}\left(\mathrm{\Gamma },\mathrm{\Lambda }\right)\equiv {\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Gamma },\mathcal{O}\right)$ and ${\mathrm{S}}_1\left(\mathrm{\Gamma },\mathrm{\Lambda }\right)\equiv {\mathrm{S}}_1\left(\mathrm{\Gamma },\mathcal{O}\right)$. Evidently $[{\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Omega },\mathrm{\Lambda }\right)\equiv {\mathit{U}}_{\mathrm{fin}}\left(\mathrm{\Lambda },\mathrm{\Lambda }\right)\equiv {\mathit{U}}_{\mathrm{fin}}\left(\mathcal{O},\mathrm{\Lambda }\right)$.

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. $F_{\sigma }$-subsets and covering properties

If $\mathcal{U}$ is a family of subsets of $X$, $A\subseteq X$, we set

$$\mathcal{U}\mid A=\left\{U\cap A:U\in \mathcal{U}\right\}.$$E12

Let $A\subseteq X$. If $U\subseteq A$ is open in the subspace topology, then there exists an open set $U^{\ast }\subseteq X$ such that $U=A\cap U^{\ast }$. Using the Axiom of Choice we choose one such set $U^{\ast }$ for each set $U\subseteq A$ open in $A$. If $\mathcal{U}$ is an open (in the subspace topology) cover of $A$ then we set ${\mathcal{U}}^{\ast }=\left\{U^{\ast }:U\in \mathcal{U}\right\}$. Then $\mathcal{U}={\mathcal{U}}^{\ast }\mid A$. Note that ${\mathcal{U}}^{\ast }$ need not be a cover of $X$.

By Theorem 3.1 of [3] we have.

Theorem 13 (W. Just, A. Miller, P. Sczeptycki and M. Scheepers [3]). Let $\mathrm{\Phi },\mathrm{\Psi }\in \left\{\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }\right\}$, $\mathrm{G}$ being one of ${\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}},{\mathit{U}}_{\mathrm{fin}}$. If a topological space $X$ possesses the covering property $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$, $F\subseteq X$ is closed, then $F$ with the subspace topology possesses the property $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ as well. Moreover, if $f:X\stackrel{\mathit{\text{onto}}}{\to }Y$ is continuous, then $Y$ possesses the property $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ as well.

Let $\mathcal{S}$ be a “topological” property of topological spaces, $\mathcal{E}\subseteq \mathcal{P}\left(X\right)$, $X\in \mathcal{E}$. We say that $\mathcal{S}$ is hereditary for $\mathcal{E}$ if assuming that $X$ has the property $\mathcal{S}$, every $A\in \mathcal{E}$ endowed with the subspace topology has the property $\mathcal{S}$ as well. If $\mathcal{E}=\mathcal{P}\left(X\right)$ then we simply say that $\mathcal{S}$ is hereditary on $X$.

By Theorem 13, for any $\mathrm{\Phi },\mathrm{\Psi }\in \left\{\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }\right\}$, $\mathrm{G}$ being one of ${\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}},{\mathit{U}}_{\mathrm{fin}}$, the covering property $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ is hereditary for closed subsets.

In [8], M. Scheepers proved that $\mathrm{add}\left({\mathrm{S}}_1\left(\mathrm{\Gamma },\mathrm{\Gamma }\right)\right)\ge \mathfrak{h}$. Since $\mathfrak{h}>{\aleph }_0$, by Theorem 13 we obtain.

Corollary 14. The covering property ${\mathrm{S}}_1\left(\mathrm{\Gamma },\mathrm{\Gamma }\right)$ is hereditary for $F_{\sigma }$-subsets of $X$.

We prove the following result.

Theorem 15. Let $\mathrm{\Phi }\in \left\{\mathcal{O},\mathrm{\Omega },\mathrm{\Gamma }\right\}$ and $\mathrm{\Psi }\in \left\{\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega }\right\}$. Let $\mathrm{G}$ be one of ${\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}},{\mathit{U}}_{\mathrm{fin}}$. The covering property $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ is hereditary for $F_{\sigma }$-subsets of $X$.

Proof: Let $F={\bigcup }_n{F}_n$, where for each $n$ the set $F_n$ is closed and $F_n\subseteq F_{n+1}$. Let the countable sequence of open $\phi$-covers of $F$ be bijectively enumerated as

$$⟨{\mathcal{U}}_{n,m}:n,m\in \omega ⟩.$$E13

Then ${\mathcal{U}}_{n,m}\mid F_n$ is a $\phi$-cover of $F_n$. Apply ${\mathrm{S}}_1\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ to the sequence $⟨{\mathcal{U}}_{n,m}|F_n:m\in \omega ⟩$ for every $n$. We obtain sequences $⟨V_{n,m}\in {\mathcal{U}}_{n,m}|F_n:m\in \omega ⟩$ such that every family $\left\{V_{n,m}:m\in \omega \right\}$ is a $\psi$-cover of $F_n$. Let $U_{n,m}\in {\mathcal{U}}_{n,m}$ be such that $V_{n,m}=U_{n,m}\cap F_n$.

One can easily see that the family $\left\{U_{n,m}:n,m\in \omega \right\}$ is a $\psi$-cover of $F$. Since $U_{n,m}\in {\mathcal{U}}_{n,m}$ for every $n$ and $m$, we obtain that $F$ possesses the covering property ${\mathrm{S}}_1\left(\mathrm{\Phi },\mathrm{\Psi }\right)$.

For $\mathrm{G}={\mathrm{S}}_{\mathrm{fin}},{\mathit{U}}_{\mathrm{fin}}$ the proof is similar.□

8. $\sigma$-space

A topological space $X$ is said to be a $\sigma$-space, if every $G_{\delta }$-subset of $X$ is an $F_{\sigma }$-set. Consequently, every Borel subset of a $\sigma$-space is an $F_{\sigma }$-set.

J. Haleš [4] proved.

Theorem 16. Let $X$ be a perfectly normal topological ${\mathrm{S}}_1\left(\mathrm{\Gamma },\mathrm{\Gamma }\right)$-space. The covering property ${\mathrm{S}}_1\left(\mathrm{\Gamma },\mathrm{\Gamma }\right)$ is hereditary on $X$ if and only if $X$ is a $\sigma$-space.

We obtain an easy Corollary.

Corollary 17. Let $\mathrm{\Phi }$ be one of the symbols $\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }$. Let $\mathrm{G}$ be one of ${\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}}$. Assume that $X$ is a perfectly normal topological space. If $X$ possesses the covering property $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Gamma }\right)$ and $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Gamma }\right)$ is hereditary on $X$, then $X$ is a $\sigma$-space.

Proof: Note that

$$\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Gamma }\right)\to \mathrm{G}\left(\mathrm{\Gamma },\mathrm{\Gamma }\right)\to {\mathrm{S}}_1\left(\mathrm{\Gamma },\mathrm{\Gamma }\right),$$E14

(if $\mathrm{G}={\mathrm{S}}_{\mathrm{fin}}$ use (9)) and $\mathrm{\Gamma }\subseteq \mathrm{\Phi }$. □

Following Haleš’ proof of Theorem 16 we obtain.

Theorem 18. Let $X$ be a $\sigma$-space, $\mathrm{\Phi }$ being one of the symbols $\mathcal{O},{\mathrm{\Lambda }}_{\mathit{\text{ctbl}}},\mathrm{\Gamma }$ and $\mathrm{\Psi }$ being one of $\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }$. Let $\mathrm{G}$ be one of ${\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}},{\mathit{U}}_{\mathrm{fin}}$. Then $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$ is hereditary on $X$.

Proof: Assume that $X$ is a topological $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Psi }\right)$-space, where $\mathrm{\Phi }=\mathcal{O},{\mathrm{\Lambda }}_{\mathit{\text{ctbl}}},\mathrm{\Gamma }$ and $\mathrm{\Psi }=\mathcal{O},\mathrm{\Lambda },\mathrm{\Omega },\mathrm{\Gamma }$. Assume also that $X$ is a $\sigma$-space and $A\subseteq X$. Let $⟨{\mathcal{U}}_n:n\in \omega ⟩$ be a sequence of open (in the subspace topology) $\phi$-covers of $A$.

If $\mathrm{\Phi }=\mathcal{O}$ then ${\mathcal{U}}_n^{\ast }\mid B$ is an $o$-cover of $B={\bigcap }_n\bigcup {\mathcal{U}}_n^{\ast }\supseteq A$. Since $B$ is a Borel set, therefore an $F_{\sigma }$-set in $X$, by Theorem 15, the set $B$ possesses the covering property $\mathrm{G}\left(\mathcal{O},\mathrm{\Psi }\right)$. So, for each $n\in \omega$ there exists a $U_n\in {\mathcal{U}}_n^{\ast }\mid B$ or finite set ${\mathcal{W}}_n\subseteq {\mathcal{U}}_n^{\ast }\mid B$ such that $\left\{U_n:n\in \omega \right\}$, or ${\bigcup }_n{\mathcal{W}}_n$, or $\left\{\bigcup {\mathcal{W}}_n:n\in \omega \right\}$ is a $\psi$-cover of $B$, respectively. Then $\left\{U_n:n\in \omega \right\}\mid A$, or ${\bigcup }_n{\mathcal{W}}_n\mid A$, or $\left\{\bigcup {\mathcal{W}}_n:n\in \omega \right\}\mid A$ is an open $\psi$-cover of $A$, respectively.

Let $\mathrm{\Phi }={\mathrm{\Lambda }}_{\mathit{\text{ctbl}}}$. Since each ${\mathcal{U}}_n$ is countable, we can assume that $\left\{U_{n,m}:m\in \omega \right\}$ is a bijective enumeration of ${\mathcal{U}}_n$. The family ${\mathcal{U}}_{n,m}=\left\{U_{n,k}:k\ge m\right\}$ is a $\lambda$-cover of $A$. If we set $B={\bigcap }_{n,m}\bigcup {\mathcal{U}}_{n,m}^{\ast }\supseteq A$ then each ${\mathcal{U}}_n^{\ast }\mid B$ is a $\lambda$-cover of $B$. Continue as above.

If $\mathrm{\Phi }=\mathrm{\Gamma }$ we can assume that each ${\mathcal{U}}_n$ is countable. For every $n\in \omega$ let $\left\{U_{n,m}:m\in \omega \right\}$ be a bijective enumeration of ${\mathcal{U}}_n$. Then ${\mathcal{U}}_n^{\ast }\mid B$ is a $\gamma$-cover of $B={\bigcap }_n{\bigcup }_k{\cap }_{m\ge k}{U}_{n,m}^{\ast }\supseteq A$. The set $B$ is Borel, therefore $F_{\sigma }$. By Theorem 15, $B$ is an $\mathrm{G}\left(\mathrm{\Gamma },\mathrm{\Psi }\right)$-space. Continue as above.□

For $\mathrm{\Phi }=\mathrm{\Psi }=\mathrm{\Gamma }$, $\mathrm{G}={\mathrm{S}}_1$, we obtain one implication of the Haleš’ Theorem 16.

Corollary 19. Let $\mathrm{\Phi }$ be one of the symbols $\mathcal{O},{\mathrm{\Lambda }}_{\mathit{\text{ctbl}}},\mathrm{\Gamma }$. Let $\mathrm{G}$ be one of ${\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}}$. The covering property $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Gamma }\right)$ is hereditary on a perfectly normal topological space $X$ if and only if $X$ is a $\sigma$-space.

Corollary 20. Let $\mathrm{\Phi }$ be one of the symbols $\mathcal{O},{\mathrm{\Lambda }}_{\mathit{\text{ctbl}}},\mathrm{\Gamma }$. Let $\mathrm{G}$ be one of ${\mathrm{S}}_1,{\mathrm{S}}_{\mathrm{fin}}$. Assume that a perfectly normal topological space $X$ possesses the covering property $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Gamma }\right)$. Then $X$ is hereditary $\mathrm{G}\left(\mathrm{\Phi },\mathrm{\Gamma }\right)$ if and only if $X$ is hereditary ${\mathrm{S}}_1\left(\mathrm{\Gamma },\mathrm{\Gamma }\right)$.

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].

Classification

2010 MSC: 54C35, 54C20, 54D55