%------------------------------------------------------------------------------ % Here please write the date of submission of paper or its revisions: %------------------------------------------------------------------------------ % \documentclass[12pt, reqno]{amsart} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, color} \usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref} \hypersetup{colorlinks=true,linkcolor=red, anchorcolor=green, citecolor=cyan, urlcolor=red, filecolor=magenta, pdftoolbar=true} %\usepackage{draftwatermark} %\usepackage{lineno} \textheight 22.5truecm \textwidth 14.5truecm \setlength{\oddsidemargin}{0.35in}\setlength{\evensidemargin}{0.35in} \setlength{\topmargin}{-.5cm} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{conjecture}[theorem]{Conjecture} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{summary}[theorem]{Summary} \newtheorem{axiom}[theorem]{Axiom} \newtheorem{problem}[theorem]{Problem} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \numberwithin{equation}{section} %%%%%%%%%%%%%%%%% %\usepackage{amssymb} %\usepackage{amsfonts} %\usepackage{cyr} \newcommand{\sone}{{\sf S}_1} \newcommand{\gone}{{\sf G}_1} \newcommand{\sfin}{{\sf S}_{fin}} \newcommand{\gfin}{{\sf G}_{fin}} \newcommand{\ufin}{{\sf U}_{fin}} \newcommand{\ssone}{{\sf S}_1^*} \newcommand{\sgone}{{\sf G}_1^*} \newcommand{\ssfin}{{\sf S}_{fin}^*} \newcommand{\sgfin}{{\sf G}_{fin}^*} %\newcommand{\sufin}{{\sf U}_{fin}^*} \newcommand{\sufin}{\ensuremath{\mathsf{U}_{fin}^*}} \newcommand{\sssone}{{\sf SS}_1^*} \newcommand{\ssgone}{{\sf SG}_1^*} \newcommand{\sssfin}{{\sf SS}_{fin}^*} \newcommand{\ssgfin}{{\sf SG}_{fin}^*} \newcommand{\naturals}{{\mathbb N}} \newcommand{\reals}{{\mathbb R}} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \usepackage{amscd} %\SetWatermarkText{Galley Proof}\SetWatermarkScale{4} \begin{document} %\linenumbers \setcounter{page}{82} \begin{center}{\footnotesize Khayyam J. Math. 1 (2015), no. 1, 82--106}\\\end{center} \noindent\parbox{2.85cm}{\includegraphics*[keepaspectratio=true,scale=0.24]{KJM.jpg}} \vspace{0.5cm} \title[Star selection principles]{Star selection principles: A survey} \author[Lj.D.R. Ko\v cinac]{Ljubi\v sa D.R. Ko\v cinac} \address{University of Ni\v s, Faculty of Sciences and Mathematics\\ 18000 Ni\v s, Serbia} \email{lkocinac@gmail.com} \dedicatory{{\rm Communicated by H.R. Ebrahimi Vishki}} \subjclass[2010]{Primary 54D20; Secondary 54A35, 54B20, 54E15, 54H10, 91A44.} \keywords{Star selection principles, {\sf ASSM}, selectively $(a)$, uniform selection principles.} \date{Received: 29 November 2014; Accepted: 30 December 2014.} %\newline \indent $^{*}$ Corresponding author} \begin{abstract} We review selected results obtained in the last fifteen years on star selection principles in topology, an important subfield of the field of selection principles theory. The results which we discuss concern also uniform structures and, in particular, topological groups and their generalizations. \end{abstract} \maketitle %%%%%%%%%%%%% 11111 %%%%%%%%%%%%%%% \section{Introduction} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% There are many results in the literature which show that a number of topological properties can be characterized by using the method of stars. In particular it is the case with many covering properties of topological spaces. The method of stars has been used to study the problem of metrization of topological spaces, and for definitions of several important classical topological notions. More information on star covering properties can be found in \cite{vDRRT}, \cite{misha-survey}. We use here such a method in investigation of selection principles for topological and uniform spaces. Although Selection Principles Theory is a field of mathematics having a rich history going back to the papers by Borel, Menger, Hurewicz, Rothberger, Seirpi\'nski in 1920--1930's, a systematic investigation in this area rapidly increased and attracted a big number of mathematicians in the last two-three decades after Scheeper's paper \cite{coc1}. Nowadays, this theory has deep connections with many branches of mathematics such as Set theory and General topology, Game theory, Ramsey theory, Function spaces and hyperspaces, Cardinal invariants, Dimension theory, Uniform structures, Topological groups and relatives, Karamata theory. Researchers working in this area have organized four international mathematical forums called ``Workshop on Coverings, Selections and Games in Topology". There are several survey papers about selection principles theory (see, for example, \cite{koc-iransurvey, koc-proc-steklov, sakai-marion} and the paper \cite{boaz-problems} for open problems). \medskip Two basic ideas in this theory are simple and may be described by the following two schemes: \smallskip \noindent {\bf Scheme 1:} To a topological property $\mathcal P$ associate selectively $\mathcal P$ as follows: $\mathcal P$: for each $A$ there is a $B$ such that ... ${\rm selectively} \mathcal P$: For each sequence $\langle A_n:n\in \naturals \rangle$ there is a sequence $\langle B_n:n\in \naturals\rangle$ such that ... \medskip \noindent {\bf Scheme 2:} $\mathcal A$ and $\mathcal B$ are given collections, $\pi$ is a procedure of selection. Apply $\pi$ to $\mathcal A$ to arrive to $\mathcal B$. \medskip For example, if $\mathcal P$ is compactness (for each open cover $\mathcal U$ of a space $X$ there is a finite subcover $\mathcal V$), then selectively $\mathcal P$ is defined as follows: for each sequence $\langle \mathcal U_n;n\in \naturals\rangle$ of open covers of $X$ there is a sequence $\langle \mathcal V_n:n\in \naturals\rangle$ of finite sets with $\mathcal V_n \subset \mathcal U_n$, $n\in\mathbb N$, and $\bigcup_{n\in\mathbb N}\mathcal V_n$ covers $X$. This property is called the Menger property (see below). Many other selective versions of classical topological concepts have been defined in this way. \medskip Three classical selection principles defined in general forms in \cite{coc1} are: \medskip Let ${\mathcal A}$ and ${\mathcal B}$ be sets consisting of families of subsets of an infinite set $X$. Then the following selection hypothesis are defined: \smallskip \noindent $\sfin({\mathcal A},{\mathcal B})$: for each sequence $\langle A_n:n\in\naturals\rangle$ of elements of ${\mathcal A}$ there is a sequence $\langle B_n:n\in\naturals\rangle$ of finite sets such that for each $n$, $B_n\subset A_n$, and $\bigcup_{n\in\naturals}B_n\in {\mathcal B}$. \smallskip \noindent $\sone({\mathcal A},{\mathcal B})$: for each sequence $\langle A_n:n\in\naturals\rangle$ of elements of ${\mathcal A}$ there is a sequence $\langle b_n:n\in\naturals\rangle $ such that for each $n$, $b_n\in A_n$, and $\{b_n:n\in\naturals\}$ is an element of ${\mathcal B}$. \smallskip \noindent $\ufin({\mathcal A},{\mathcal B})$: for each sequence $\langle A_n:n\in\naturals\rangle$ of elements of ${\mathcal A}$ there is a sequence $\langle B_n:n\in\naturals\rangle $ such that for each $n$, $B_n$ is a finite subset of $A_n$ and $\{\bigcup B_n:n\in\naturals\} \in \mathcal B$. \medskip In this paper we use the following notation for collections of covers of a topological space $X$: \begin{itemize} \item $\mathcal O$ is the collection of all open covers of $X$; \item $\Omega$ is the collection of $\omega$-covers of $X$. An open cover $\mathcal U$ of $X$ is said to be an \emph{$\omega$-cover} if each finite subset of $X$ is contained in a member of $\mathcal U$ and $X\notin \mathcal U$; \item $\Gamma$ denotes the collection of $\gamma$-covers of $X$. An open cover $\mathcal U$ of $X$ is said to be a \emph{$\gamma$-cover} if each point of $X$ does not belong to at most finitely many elements of $\mathcal U$. \end{itemize} Then: \begin{itemize} \item[{\sf M}:] $\sfin(\mathcal O,\mathcal O)$ is the \emph{Menger property} \cite{menger}, \cite{hurewicz}; \item[{\sf R}:] $\sone(\mathcal O,\mathcal O)$ is the \emph{Rothberger property} \cite{rothberger}; \item[{\sf H}:] $\ufin(\Gamma,\Gamma)$ is the \emph{Hurewicz property} \cite{hurewicz} \end{itemize} \medskip The paper is organized in the following way. Immediately after this introduction in Section 2 we give information about terminology and notation, and also about known topological constructions we use in this paper. In Section 3 we discuss in details star selection principles in topological spaces. The next two sections are devoted to neighbourhood and absolute star selection properties, two variations of the properties we considered in Section 3. In particular, in Subsection 5.2 we report results on selectively $(a)$ spaces. In the second part of the paper we turn attention to appearance of star selection properties in special classes of topological structures: uniform and quasi-uniform spaces, and, especially, in topological and paratopological groups. Each section contains some open problems which can motivate new researches for work in this field. %%%%%%% 22222 %%%%%%%%%%%%%%%%%%%%%%% \section{Definitions and terminology} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \medskip Throughout the paper ``space" means ``topological space". By $\mathbb N$, $\mathbb Z$, and $\mathbb R$ we denote the set of natural numbers, integers, and real numbers, respectively. The symbol $\omega$ denotes the set of nonnegative integers and also the first infinite ordinal, while $\omega_1$ is the first uncountable ordinal. The cardinality of continuum is denoted by $\mathfrak c$, and {\sf CH} denotes the Continuum Hypothesis. Most of undefined notations and terminology are as in \cite{engelking}. If $X$ is a space, $\mathcal K$ a collection of subsets of $X$, $A$ a subset of $X$, and $x\in X$, then ${\rm St}(A,\mathcal K)$ is the union of all elements in $\mathcal K$ which meet $A$. We write ${\rm St}(x,\mathcal K)$ instead of ${\rm St}(\{x\},\mathcal K)$. \medskip We recall known topological constructions which will be used in next sections without special mention. \smallskip {\bf A.} ({\bf The Baire space $^{\omega}\omega$}) \, Let $^{\omega}\omega$ be the set of all functions $f:\omega \to \omega$ (in fact, the countable Tychonoff power of the discrete space $D(\omega)$). A natural pre-order $\prec\sp *$ on $^{\omega}\omega$ is defined by $f\prec\sp * g$ if and only if $f(n) \le g(n)$ for all but finitely many $n$. A subset $F$ of $^{\omega}\omega$ is said to be \emph{dominating} if for each $g\in\,^{\omega}\omega$ there is a function $f\in F$ such that $g\prec\sp * f$. A subset $F$ of $^{\omega}\omega$ is called \emph{bounded} if there is an $g\in\,^{\omega}\omega$ such that $f\prec\sp * g$ for each $f\in F$. The symbol $\mathfrak b$ (resp. $\mathfrak d$) denotes the least cardinality of an unbounded (resp. dominating) subset of $^{\omega}\omega$. Another uncountable small cardinal characterized (by Bartoszy\'nski in 1987) in terms of subsets of $^{\omega}\omega$ is the cardinal ${\sf cov}(\mathcal M)$, the \emph{covering number of the ideal of meager subsets of $\mathbb R$}: \[ {\sf cov}(\mathcal M) = \min\{|F|:F\subset {^\omega}{\omega} \mbox{ such that } \forall g\in {^\omega}{\omega} \ \exists f\in F \mbox{ with } f(n)\neq g(n) \forall n\in\omega\}. \] Recommended literature concerning uncountable small cardinals is \cite{vanDouwen-handbook} and \cite{vaughan}. \medskip {\bf B.} ({\bf $\Psi$-spaces}) \, A family $\mathcal A$ of infinite subsets of $\naturals$ is called \emph{almost disjoint} if the intersection of any two distinct sets in $\mathcal A$ is finite. Let $\mathcal A$ be an almost disjoint family. The symbol $\Psi (\mathcal A)$ denotes the space $\naturals \cup \mathcal A$ with the following topology: all points of $\naturals$ are isolated; a basic neighborhood of a point $A$ in $\mathcal A$ is of the form $\{A\} \cup (\naturals \setminus F)$, where $F$ is a finite subset of $\naturals$. \medskip {\bf C.} ({\bf Pixley-Roy space}) \, For a space $X$, let ${\sf PR}(X)$ be the space of all nonempty finite subsets of $X$ with the Pixley-Roy topology \cite{vanDouwen-PR}: for $A \in �{\sf PR}(X)$ and an open set $U \subset X$, let $[A,U] = \{B \in {\sf PR}(X) : A \subset B \subset U\}$; the family $\{[A,U] : A \in�{\sf PR}(X), U \mbox{ open in } X\}$ is a base for the Pixley-Roy topology. Obviously $\{\{x\} : x \in X\}$ is closed and discrete in ${\sf PR}(X)$. Therefore, ${\sf PR}(X)$ is Lindel\"of if and only if $X$ is countable. It is known that (1) for a $T_1$-space $X$, ${\sf PR}(X)$ is always zero-dimensional, Tychonoff and hereditarily metacompact, and (2) ${\sf PR}(X)$ is developable if and only if $X$ is first-countable (see \cite{vanDouwen-PR}). \medskip {\bf D.} ({\bf Alexandroff duplicate}) \, Let $(X,\tau)$ be a topological space. The Alexandroff duplicate of $X$ (see \cite{engelking}, \cite{agata-watson}) is the set ${\sf AD}(X):= X\times \{0,1\}$ equipped with the following topology. For each $U\in \tau$ let $\widehat{U} = U\times \{0,1\}$. Define a base for a topology on ${\sf AD(X)}$ by $\mathcal B = \mathcal B_0 \cup \mathcal B_1$, where $\mathcal B_0$ is the family of all sets $\widehat{U} \setminus (F\times \{1\}) \subset {\sf AD}(X)$, with $U\in \tau$ and $F$ a finite subset of $X$, and $\mathcal B_1 =\{\langle x,1\rangle :x\in X\}$. For every $x\in X$ put $\tau_x= \{U\in \tau: x\in U\}$ and $\mathcal B_{\langle x,0\rangle } = \{\widehat{U} \setminus \{\langle x,1\rangle \}:U\in \tau_x\}$, and $\mathcal B_{\langle x,1\rangle} = \{\{\langle x,1\rangle \}\}$. Then, if $X$ is a $T_1$-space, $\mathcal B_{\langle x,0\rangle }$ is a local base at each $\langle x,0\rangle \in {\sf AD}(X)$, and $\mathcal B^{\prime} = \bigcup_{x\in X}(\mathcal B_{\langle x,0\rangle } \cup B_{\langle x,1\rangle })$ is a base in ${\sf AD}(X)$ such that $\mathcal B^{\prime} \subset \mathcal B$. If $\mathcal U$ is a family of open sets in $X$, then we say that the family $\mathcal U^{\ast}:= \{\widehat{U} \setminus (F\times \{1\}):U\in \mathcal U, F \mbox{ a finite subset of } X\}$ of open subsets of ${\sf AD}(X)$ is associated to $\mathcal U$ and vice versa. \smallskip For many topological properties $\mathcal P$ the space ${\sf AD}(X)$ has $\mathcal P$ if $X$ has $\mathcal P$ (see, for example, \cite{agata-watson}). Such properties are, for instance, complete regularity, normality, compactness, Lindel\"ofness, (hereditary) paracompactness. \smallskip Recall also the definition of subspaces (called lines) of ${\sf AD}(X)$. Let $A$ and $B$ be disjoint subspaces of $X$. The subspace $Z= (A\times \{1\}) \cup (B\times \{0\})$ of ${\sf AD}(X)$ is called a \emph{Michael-type line} (see \cite[Definition 3.14]{agata-watson}). %%%%%%%%%%%%%%%%%%%%%% 33333 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Star selection principles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In \cite{fle} it was proved that a Hausdorff space $X$ is countably compact if and only if for every open cover $\mathcal U$ of $X$ there exists a finite subset $F\subset X$ such that ${\rm St}(F,\mathcal U)=X$. This result was a motivation for the following two definitions that appeared in \cite{vDRRT}. A space $X$ is \emph{starcompact} if for every open cover $\mathcal U$ of $X$ there exists a finite subset $\mathcal V$ of $\mathcal U$ such that ${\rm St}(\cup \mathcal V,\mathcal U)=X$. A space $X$ is \emph{strongly starcompact} if for every open cover $\mathcal U$ of $X$ there exists a finite subset $F\subset X$ such that ${\rm St}(F,\mathcal U)=X$. \medskip Applying now Schemes 1 and 2 we define selective versions of these notions, and modifying them we obtain the following star selection principles introduced by the author of this article in \cite{koc-starmenger} (see also \cite{koc-starmenger2}). \medskip Let ${\mathcal O}$ be the collection of all open covers of a space $X$, ${\mathcal B}$ a subcollection of $\mathcal O$, and $\mathcal K$ a family of subsets of $X$. Then: \smallskip \noindent {\bf 1.} The symbol $\ssfin(\mathcal O,\mathcal B)$ denotes the selection hypothesis: \begin{quote} For each sequence $\langle \mathcal U_n:n\in\naturals\rangle $ of elements of $\mathcal O$ there is a sequence $\langle \mathcal V_n:n \in \naturals\rangle $ such that for each $n\in \naturals$, $\mathcal V_n$ is a finite subset of $\mathcal U_n$, and $\{{\rm St}(\cup \mathcal V_n,\mathcal U_n):n \in \naturals\} \in \mathcal B$; \end{quote} \noindent {\bf 2.} The symbol $\ssone(\mathcal O,\mathcal B)$ denotes the selection hypothesis: \begin{quote} For each sequence $\langle \mathcal U_n:n\in\naturals\rangle $ of elements of $\mathcal O$ there is a sequence $\langle U_n:n \in \naturals\rangle $ such that for each $n\in \naturals$, $U_n \in\mathcal U_n$ and $\{{\rm St}(U_n,\mathcal U_n):n\in\naturals\} \in \mathcal B$; \end{quote} \noindent {\bf 3.} ${\sf SS}_{\mathcal K}^{*}(\mathcal O,\mathcal B)$ denotes the selection hypothesis: \begin{quote} For each sequence $\langle \mathcal U_n:n\in \naturals\rangle $ of elements of $\mathcal O$ there exists a sequence $\langle K_n:n \in \naturals\rangle $ of elements of $\mathcal K$ such that $\{{\rm St}(K_n, \mathcal U_n):n \in \naturals\} \in\mathcal B$. \end{quote} \noindent When $\mathcal K$ is the collection of all finite (resp. one-point, compact) subspaces of $X$ we write $\sssfin(\mathcal O,\mathcal B)$ (resp., $\sssone(\mathcal O,\mathcal B)$, ${\sf SS}_{{\rm K}}^{*}(\mathcal O,\mathcal B)$) instead of ${\sf SS}_{\mathcal K}^{*}(\mathcal O,\mathcal B)$. \medskip The following terminology we borrow from \cite{koc-starmenger}. \smallskip \noindent For a space $X$ we have: \noindent \ \ {\sf SM}: the \emph{star-Menger property} = $\ssfin(\mathcal O,\mathcal O)$; \noindent \ \ {\sf SR}: the \emph{star-Rothberger property} = $\ssone(\mathcal O,\mathcal O)$; \noindent \ {\sf SSM}: the \emph{strongly star-Menger property} = $\sssfin(\mathcal O,\mathcal O)$; \noindent \ {\sf SSR}: the \emph{strongly star-Rothberger property} = $\sssone(\mathcal O,\mathcal O)$; \noindent {\sf SS-K-M}: the \emph{star-$K$-Menger property} = ${\sf SS}_{{\rm K}}^{*}(\mathcal O,\mathcal O)$. \medskip In \cite{star-hur}, two star versions of the Hurewicz property were studied: \smallskip \noindent \ \ {\sf SH}: the \emph{star-Hurewicz property} = $\ssfin(\mathcal O,\Gamma)$; \noindent \ {\sf SSH}: the \emph{strongly star-Hurewicz property} = $\sssfin(\mathcal O,\Gamma)$. \medskip It is clear that each of properties {\sf SM}, {\sf SH}, {\sf SR} can be viewed as a selective version of starcompactnes, while the properties {\sf SSM}, {\sf SSH}, {\sf SSR}, {\sf SS-K-M} can be viewed as selective versions of strong starcompactness. Starcompctness implies {\sf SH}, hence also {\sf SM}, and strong starcompactness implies {\sf SSH} and thus {\sf SSM}. In \cite[Example 2.3]{koc-starmenger} we have shown that the Tychonoff Plank $[0,\omega_1]\times [0,\omega] \setminus \{\langle\omega_1,\omega\rangle \}$ is {\sf SSM} but not strongly starcompact. On the other hand, in \cite[Example 2.1]{song-polish} it is proved that the Tychonoff Plank is {\sf SSH} but not starcompact (thus not strongly starcompact). It is worth to mention that for each ordinal $\alpha$, the space $[0,\alpha)$ with the order topology is ${\sf SSR}$. Of course, Menger spaces are {\sf SSM}, and every {\sf SSM} space is {\sf SM}. Similarly for the Hurewicz and Rothberger properties. The simplest example which shows that the converse need not be true is the ordinal space $[0,\omega_1)$ which is ${\sf SSH}$ (hence {\sf SSM}, {\sf SH}, {\sf SM}) but not {\sf M} (thus not {\sf H}) (see \cite{koc-starmenger} and \cite{star-hur}). By results in \cite{koc-starmenger} and \cite{star-hur} we have that every metacompact (every open cover $\mathcal U$ has a point-finite open refinement $\mathcal V$) strongly star-Menger space is Menger, and that for paracompact Hausdorff spaces the three Menger-type properties, ${\sf SM}$, ${\sf SSM}$ and ${\sf M}$ are equivalent \cite{koc-starmenger}. The same situation is with the classes ${\sf SSR}$, ${\sf SR}$ and ${\sf R}$ \cite{koc-starmenger} and ${\sf SSH}$, ${\sf SH}$ and ${\sf H}$ \cite{star-hur}. Let us mention the following \begin{example} \rm (\cite[Example 2.2]{song-polish})\label{sh-not-ssh} There is a Tychonoff ${\sf SH}$ space which is not ${\sf SSH}$. \smallskip Such a space is $\alpha D(\mathfrak{c}) \times [0,\mathfrak{c}^+] \setminus \{\langle\infty, \mathfrak{c}^+\rangle\}$ of the product $\alpha D(\mathfrak{c}) \times [0,\mathfrak{c}^+]$, where $\alpha D(\mathfrak{c}) = D(\mathfrak{c})\cup\{\infty\}$ is the one-point compactification of the discrete space $D(\mathfrak{c})$ of cardinality $\mathfrak{c}$. \end{example} \medskip Following the general definition of ${\sf SS}^{*}_{\mathcal K}(O,O)$ (the beginning of this section) and taking $\mathcal K$ to be the collection of countably compact spaces Song defined star-$C$-Menger spaces in \cite{song-c-menger} (he also studied star-$K$-Menger spaces in \cite{song-star-k-menger}). He proved: \begin{example} \rm (\cite[Example 2.2]{song-c-menger}) There exists a Tychonoff star-$C$-Menger space which is not star-$K$-Menger. \end{example} \smallskip Now we are going to see how above mentioned star selection properties are related to $\Psi$-spaces and Pixley-Roy spaces. In fact, in $\Psi$-spaces $\Psi (\mathcal A) = \omega \cup \mathcal A$ star selection properties strongly depend on the cardinality of the almost disjoint family $\mathcal A$ and are related to small infinite cardinals. The first results of this kind appeared in the preprint/draft \cite{matveev-preprint} sent me by the author in July 1998 (see \cite[Example 2.2]{koc-starmenger} and \cite{star-hur}), and then included in the paper \cite{misha-milena-psi}. By combining the results from \cite{matveev-preprint} and \cite{misha-milena-psi} we can formulate the following \begin{theorem} The following hold for a $\Psi$-space $\Psi(\mathcal A)$: \begin{itemize} \item[$(1)$] $\Psi(\mathcal A)$ is {\sf SSM} if and only if $|\mathcal A| < \mathfrak d$. If $|\mathcal A|=\mathfrak c$, then $\Psi(\mathcal A)$ is not {\sf SM}, and if $\mathcal A| < \aleph_{\omega}$, then $\Psi(\mathcal A)$ is {\sf SM} if and only if it is {\sf SSM}; \item[$(2)$] $\Psi(\mathcal A)$ is {\sf SSH} if and only if $|\mathcal A| < \mathfrak b$; \item[$(3)$] If $|\mathcal A| < {\sf cov}(\mathcal M)$, then $\Psi(\mathcal A)$ is {\sf SSR}. There is an almost disjoint family $\mathcal A$ of cardinality ${\sf cov}(\mathcal M)$ such that $\Psi(\mathcal A)$ is not {\sf SSR}. \end{itemize} \end{theorem} In \cite{sakai-SM}, Sakai investigated star-Mengerness in the Pixley-Roy space. He established the following: \begin{theorem} $(1)$ If ${\sf PR}(X)$ is star-Menger, then $|X| < \mathfrak{c}$ holds. Hence, under ${\sf CH}$, ${\sf PR}(X)$ is star-Menger if and only if $X$ is countable; $(2)$ If ${\sf PR}(X)$ is star-Menger, then every finite power of $X$ is Menger. $(3)$ If $X$ is a cosmic space of cardinality less than $\mathfrak{d}$, then every finite power of ${\sf PR}(X)$ is star-Menger; $(4)$ Let $X$ be a semi-stratifiable space {\rm \cite{creede}}. If \ ${\sf PR}(X)$ is star-Menger, then ${\sf PR}(X)^{\kappa}$ is weakly Menger for any cardinal $\kappa$; $(5)$ If $X$ is first-countable and ${\sf PR}(X)$ is star-Menger, then ${\sf PR}(X)$ is weakly Menger. \end{theorem} A space $X$ is said to be \emph{weakly Menger} \cite{daniels} if for each sequence $\langle \mathcal U_n:n\in\naturals\rangle$ of open covers of $X$ there is a sequence $\langle \mathcal V_n:n\in\naturals\rangle$ of finite sets such that for each $n$, $\mathcal V_n\subset \mathcal U_n$ and $\overline{\bigcup_{n\in\naturals}\bigcup \mathcal V_n} = X$. \bigskip Since the very beginning of the theory of star selection principles one the following question was one of the most interesting: how large the extent of {\sf SM} or {\sf SSM} spaces can be. Recall that the \emph{extent} $e(X)$ of a space $X$ is the supremum of cardinalities of closed discrete subspaces of $X$. Recently, some results in this connection have been obtained by Y.-K. Song \cite{song-cmuc} and M. Sakai \cite{sakai-SM}, and also by B. Tsaban \cite{boaz-arxiv}. Song \cite[Example 2.4]{song-cmuc} observed that the extent of a $T_1$ strongly star-Menger space can be arbitrarily large, and asked whether there is a Tychonoff strongly star-Menger space $X$ such that $e(X) \ge \mathfrak{c}$. Answering this question, Sakai proved in \cite[Corollaries 2.2, 2.6]{sakai-SM}: \begin{theorem} $(1)$ The extent of a regular strongly star-Menger space cannot exceed $\mathfrak{c}$; $(2)$ If $X$ is a star-Menger space with $w(X) = \mathfrak{c}$, then every closed and discrete subspace of X has cardinality less than $\mathfrak{c}$; $(3)$ Let $X$ be a normal star-Menger space. Then $e(X) \le \mathfrak{d}$; $(4)$ The assertion every developable strongly star-Menger space is separable and metrizable is equivalent to $\omega_1 = \mathfrak{d}$; $(5)$ The statements $\omega_1 = \mathfrak{d}$ is equivalent to the statement that for every strongly star-Menger space $X$, $e(X) \le \omega$ holds. \end{theorem} The following problem was posed by Sakai. \begin{problem} {\rm (\cite[Question 3.3]{sakai-SM})} Can the extent of a metacompact (or, subparacompact) star-Menger space be arbitrarily large \end{problem} \medskip Another interesting question regarding star selection principles is their relations with the Alexandroff double. Some of results in this direction are listed below. 1. (\cite[Corollary 2.9]{song-cmuc}) If $X$ is an {\sf SSM} $T_1$-space, then ${\sf AD}(X)$ is {\sf SSM} if and only if $e(X) < \omega_1$. 2. It was observed in \cite{song-polish} that the Alexandroff double of the {\sf SH} space in Example \ref{sh-not-ssh} is not {\sf SH}. 3. (\cite[Theorem 2.4]{song-ssh}) If $X$ is a $T_1$-space and ${\sf AD}(X)$ is an {\sf SSH} space, then $e(X) < \omega_1$. The last result suggests the following problem. \begin{problem} Is the Alexandorff duplicate ${\sf AD}(X)$ of an {\sf SSH} space $X$ with $e(X) < \omega_1$ also {\sf SSH} \end{problem} \subsection{Operations} Most of star selection properties are not hereditary. Even more, they are not preserved by nice subspaces such as (regular) closed. It was proved for {\sf SM} and {\sf SSM} spaces in \cite{song-houston-sm}, for {\sf SH} and {\sf SSH} spaces in \cite{song-li}, and for {\sf SR} and {\sf SSR} spaces in \cite{song-sroth}. Let us formulate once again a still open question from \cite{koc-starmenger}. \begin{problem} Characterize hereditarily {\sf SM} ({\sf SSM}, {\sf SR}, {\sf SSR}, {\sf SH}, {\sf SSH}) spaces. \end{problem} There are some partial answers to this question. For example, {\sf SSM} and {\sf SSH} spaces are preserved by open $F_{\sigma}$-sets (see \cite{song-houston-sm} and \cite{song-li}, respectively), while {\sf SSR} property is preserved by clopen subspaces \cite{song-sroth}. \medskip It is known and easy to prove that continuous mappings preserve {\sf SSM}, {\sf SH}, and {\sf SSH}) spaces (see \cite{song-cmuc}, \cite{song-polish}, \cite{song-ssh}, respectively). Open and closed finite-to-one mappings preserve {\sf SSM} and {\sf SSH} spaces (\cite{song-cmuc} and \cite{song-ssh}) in the preimage direction, while open, perfect mappings preserve {\sf SH} spaces in the preimage direction \cite{song-polish}. On the other hand, it was proved in \cite{song-ssh} that assuming $\mathfrak b = \mathfrak c$ and �$\neg {\sf CH}$, there exists a closed 2-to-1 continuous mapping $f : X �\to Y$ such that $Y$ is {\sf SSH}, but $X$ is not. \medskip The product of two {\sf SM} (resp. {\sf SH}) spaces need not be in the same class. For {\sf SSH} spaces, for instance, it was shown in \cite{song-ssh}. But if one factor is compact, then the product is in the same class \cite{koc-starmenger}, \cite{star-hur}. Similarly, the product a star-$C$-Menger space and a compact space is also star-$C$-Menger \cite{song-c-menger}. However, under $\mathfrak b = \mathfrak c$ and $\neg CH$, there exist an {\sf SSH} space $X$ and a compact space $Y$ such that $X \times Y$ is not {\sf SSH} \cite{song-ssh}. Let us observe that a Lindel\"of space is not a preserving factor for the classes ${\sf SSM}$ and ${\sf SSH}$ \cite{koc-starmenger, star-hur}. The following question is an open problem. \begin{problem} {\rm (\cite{song-ssh})} Do there exist a {\sf ZFC} example of an {\sf SSH} space $X$ and a compact space $Y$ such that $X\times Y$ is not {\sf SSH} \end{problem} In \cite{koc-starmenger} we posed the following still open problem. \begin{problem} Characterize spaces $X$ which are {\sf SM} ({\sf SSM}, {\sf SR}, {\sf SSR}) in all finite powers. \end{problem} A partial solution of this problem was given in \cite{star-hur}. \begin{theorem}\label{smpowers} The following statements hold: \begin{itemize} \item[$(1)$] If each finite power of a space $X$ is ${\sf SM}$, then $X$ satisfies $\ssfin(\mathcal O,\Omega)$; \item[$(2)$] If all finite powers of a space $X$ are {\sf SSM}, then $X$ satisfies $\sssfin(\mathcal O,\Omega)$. \end{itemize} \end{theorem} In the same paper we have the following two assertions. (We remind the reader that the symbol $\mathcal O^{wgp}$ denotes the collection of weakly groupable covers of a space. A countable open cover $\mathcal U$ of a space $X$ is said to be \emph{weakly groupable} if there is a partition $\mathcal U = \bigcup_{n\in\mathbb N}\mathcal U_n$ of $\mathcal U_n$ into finite, pairwise disjoint subcollections, so that for each finite subset $F$ of $X$ there is $n\in \mathbb N$ with $F \subset \bigcup\mathcal U_n$.) \begin{theorem}\label{ufin-wgp} For a space $X$ the following are equivalent: \begin{itemize} \item[$(1)$] $X$ satisfies $\ssfin(\mathcal O,\Omega)$; \item[$(2)$] $X$ satisfies $\ssfin(\mathcal O,\mathcal O^{wgp})$. \end{itemize} \end{theorem} \begin{theorem}\label{ssfin-wgr} For a space $X$ the following are equivalent: \begin{enumerate} \item[$(1)$] {$X$ satisfies $\sssfin(\mathcal O,\Omega);$} \item[$(2)$] {$X$ satisfies $\sssfin(\mathcal O,\mathcal O^{wgp})$.} \end{enumerate} \end{theorem} So, we have actually the following problem. \begin{problem} Does $X \in \ssfin(\mathcal O,\mathcal O^{wgp})$ imply that all finite powers of $X$ are star-Menger Is it true that $\ssfin(\mathcal O,\Omega) = \ssfin(\mathcal O,\mathcal O^{wgp})$ Does $X \in \sssfin(\mathcal O,\mathcal O^{wgp})$ imply that each finite power of $X$ is {\sf SSM} \end{problem} The following result was proved in \cite{star-hur}. First we recall that $\mathcal O^{gp}$ denotes the collection of groupable covers of a space. A countable open cover $\mathcal U$ of a space $X$ is said to be \emph{groupable} if there is a partition $\mathcal U = \bigcup_{n\in\mathbb N}\mathcal U_n$ of $\mathcal U_n$ into finite, pairwise disjoint subcollections, so that each $x\in X$ belongs to all but finitely many $\bigcup \mathcal U_n$. \begin{theorem}\label{sshgroup} For a space $X$ the following are equivalent: \begin{enumerate} \item[$(1)$]{$X$ has the strongly star-Hurewicz property;} \item[$(2)$]{$X$ satisfies the selection principle $\sssfin(\mathcal O, \mathcal O^{gp})$.} \end{enumerate} \end{theorem} This result naturally suggests the following \begin{problem} Is it true that $\ssfin(\mathcal O,\Gamma)= \ssfin(\mathcal O,\mathcal O^{gp})$ \end{problem} Let us end this section by some comments. \medskip 1. In this paper we did not consider connections between star selection properties and games naturally associated to them. [For example, the \emph{strongly star-Hurewicz game} illustrates this situation; it is defined as follows. Let $X$ be a space. Two players, ONE and TWO, play a round per each natural number $n$. In the $n$--th round ONE chooses an open cover $\mathcal U_n$ of $X$ and TWO responds by choosing a finite set $A_n\subset X$. A play $\mathcal U_1,A_1;\cdots;\mathcal U_n,A_n;\cdots$ is won by TWO if $\{{\rm St}(A_n,\mathcal U_n):n \in \naturals\}$ is a $\gamma$-cover of $X$; otherwise, ONE wins. Evidently, if ONE has no winning strategy in the strongly star-Hurewicz game, then $X$ is an {\sf SSH} space. But the converse need not be true.] It would be interesting to study these connections for all classes we discussed in this section. \medskip 2. We also did not discuss relative versions of star selection principles (initiated by the author) that can be found in the literature (see, for instance, \cite{star-hur}, \cite{bon-pans-relative}). \medskip 3. Recently, I introduced \emph{selection principles in relator spaces} as generalizations of uniform selection principles. My PhD student Kocev studied these selection properties in \cite{darko-relator-amh}, \cite{darko-relator-filomat}, \cite{darko-doct}. We did not include these results in this survey although there are many interesting results and open questions in this connection. 4. Selection properties of fuzzy metric spaces \cite{koc-fuzzy} are a kind of star selection properties. %%%%%%%%%%%%%%%%%%% 44444 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Neighbourhood star selection principles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In this section we investigate star selection principles which are very close to the already considered star selection properties, but defined by neighbourhoods and stars. Selection properties defined in this way are weaker than the Menger, Rothberger and Hurewicz properties and are between strong star versions and star versions of the corresponding properties. The definitions of these selection principles were given in \cite[Def. 0.2]{koc-starmenger2}, and studied in details in \cite{bckm-mv}. Our exposition here mainly follows the last mentioned paper. \begin{definition} \rm Let $\mathcal O$ and $\mathcal B$ be as in the previous section. A space $X$ satisfies: \item[${\sf NSM}({\mathcal O},{\mathcal B})$] if for every sequence $\langle {\mathcal U}_n : n\in\mathbb N\rangle $ of elements of $\mathcal A$ one can choose finite $A_n \subset X$, $n\in\Bbb N$, so that for every open $O_n \supset A_n$, $n\in\mathbb N$, $\{{\rm St}(O_n,{\mathcal U}_n) : n\in \mathbb N\}\in\mathcal B$; \item [${\sf NSR}({\mathcal O},{\mathcal B})$] if for every sequence $\langle {\mathcal U}_n : n\in\mathbb N\rangle $ of elements of $\mathcal A$ one can choose $x_n \in X$, $n\in\mathbb N$, so that for every open $O_n \ni x_n$, $n\in\mathbb N$, $\{{\rm St}(O_n,{\mathcal U}_n) : n\in \mathbb N\}\in\mathcal B$; \item[${\sf NSH}({\mathcal O},{\mathcal B})$]if if for every sequence $\langle {\mathcal U}_n : n\in\mathbb N\rangle $ of elements of $\mathcal A$ one can choose finite $A_n \subset X$, $n\in\mathbb N$, so that for every open $O_n \supset A_n$, $n\in\mathbb N$, and for every $x\in X$, $x\in {\rm St}(O_n,{\mathcal U}_n)$ for all but finitely many $n$. \end{definition} In particular we have the following definitions: \begin{definition} \rm A space $X$ is: \begin{itemize} \item[${\sf NSM}$:] (\emph{neighbourhood star-Menger}) if the selection hypothesis ${\sf NSM}(\mathcal O,\mathcal O)$ is true for $X$; \item[${\sf NSR}$:] (\emph{neighbourhood star-Rothberger}) if the property ${\sf NSR}(\mathcal O,\mathcal O)$ is true for $X$; \item[${\sf NSH}$:] (\emph{neighbourhood star-Hurewicz}) if the selection hypothesis ${\sf NSH}({\mathcal O},{\Gamma})$ is true for $X$. \end{itemize} \end{definition} \medskip \noindent{\bf Note.} ${\sf NSR}$ and ${\sf NSM}$ spaces (as well as neighbourhood star-$K$-spaces) were defined in \cite{koc-starmenger2} under different names (nearly strongly star-Rothberger and nearly strongly star-Menger spaces). \begin{remark}\rm Since in the class of paracompact Hausdorff we have that ${\sf R} \Leftrightarrow {\sf SR}$, ${\sf M} \Leftrightarrow {\sf SM}$ (see \cite{koc-starmenger}) and ${\sf H} \Leftrightarrow {\sf SH}$ (see \cite{star-hur}), we have that in the class of paracompact Hausdorff spaces all Rothberger-type properties, all Menger-type properties and all Hurewicz-type properties considered are equivalent respectively (see Diagram 1). \end{remark} The implications ${\sf NSM}\Rightarrow {\sf SM}$, ${\sf NSH} \Rightarrow {\sf SH}$ and ${\sf NSR} \Rightarrow {\sf SR}$ can not be reversed as the following example shows. \begin{example} \rm (\cite[Example 3.7]{bckm-mv}) A Tychonoff space which is ${\sf SR}$ and ${\sf SH}$ (and thus {\sf SM}), but is neither of ${\sf NSR}$, ${\sf NSH}$, ${\sf NSM}$. \end{example} Such a space $X$ is constructed in the following way. Let $\kappa$ be an uncountable cardinal and $\alpha(D(\kappa)) = D(\kappa) \cup \{\infty\}$ the one point compactification of the discrete space $D(\kappa)$. Set $X_0= \alpha D(\kappa) \times [0,\kappa^+)$, $X_1=D(\kappa) \times\{\kappa^+\}$, $X=X_0\cup X_1$. Endow $X$ with the topology inherited from the product $\alpha D(\kappa)\times [0,\kappa^{+}]$. \medskip We show now that consistently, ${\sf NSM}$, ${\sf NSH}$ and {\sf NSR} do not imply ${\sf SSM}$, ${\sf SSH}$ and {\sf SSR}, respectively. \begin{example}\rm (\cite[Examples 3.1--3.3]{bckm-mv}) \label{3.1-3.3} Let $S$ be a subset of $\mathbb R$ such that $|S|=\omega_1$ and for every nonempty open $U\subset \mathbb R$, $|S\cap U| = \omega_1$. Set $X_S = S\times[0,\omega]$ topologized in the following way: (i) a basic neighbourhood of a point $\langle x,n\rangle \in X_S$ has the form $((U\cap S)\setminus A)\times \{n\}$, where $U$ is a neighbourhood of $x$ in the usual topology of $\mathbb R$ and $A$ is a countable set not containing $x$; (ii) a point $\langle x,\omega\rangle$, $x\in S$, has basic neighbourhoods of the form $((U\cap S)\setminus A)\times (n,\omega) \cup \{\langle x,\omega\rangle\}$, where $U$ is a neighbourhood of $x$ in the usual topology of $\mathbb R$, $A$ is a countable subset of $S$, and $n\in\omega$. Then $X_S$ is a Urysohn space and: \smallskip (1) Under $\omega_1 < \mathfrak d$ the space $X_S$ is an ${\sf NSM}$ space which is not ${\sf SSM}$. \smallskip (2) Under $\omega_1 < \mathfrak b$, $X_S$ is an ${\sf NSH}$ space which is not ${\sf SSH}$. \smallskip (3) Under $\omega_1 < {\sf cov}(\mathcal M)$, $X_S$ is an {\sf NSR} space which is not {\sf SSR}. \end{example} The following problem is still open. \begin{problem} {\rm (\cite[Problem 3.6]{bckm-mv})} Do there exist {\sf ZFC} examples of spaces as in Example {\rm \ref{3.1-3.3}} \end{problem} %%%%%%%%%%%%%%%% 55555 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Absolute versions of selection principles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% In \cite{matveev-acc} Matveev introduced the class of absolutely countable compact spaces: A space $X$ is \emph{absolutely countable compact} (shortly {\sf acc}) if for each open cover $\mathcal U$ of $X$ and each dense subset $D$ of $X$ there is a finite $A\subset D$ such that ${\rm St}(A,\mathcal U) = X$. In his subsequent paper \cite{matveev97}, Matveev applied a similar idea to introduce the following property: a space $X$ is said to be an \emph{$(a)$-space} if for each open cover $\mathcal U$ of $X$ and each dense subset $D$ of $X$ there is a closed discrete (in $X$) set $A\subset D$ such that ${\rm St}(A,\mathcal U) = X$. He also defined the class of $(wa)$-spaces replacing in the previous definition ``closed discrete" by ``discrete". These spaces were studied in a number of papers \cite{gauldvamana}, \cite{justmatvszept}, \cite{rudinjerry}, \cite{samuel-a}, \cite{szeptvaughan}. In 2010, we employed Matveev's idea to define selective versions of several star selection principles in the following general form (see \cite[p. 1361]{sel-(a)}). \begin{definition} \rm (\cite{sel-(a)}) Let $\mathcal O$ and $\mathcal B$ be collections of open covers of a space $X$ as mentioned above, and let $\mathcal K$ be a collection of subsets of $X$. Then $X$ is said to be a \emph{selectively $(\mathcal O,\mathcal B)$-$(a)_{\mathcal K}$-space}, denoted by $X\in {\sf Sel}(\mathcal O,\mathcal B)$-$(a)_{\mathcal K}$, if for each sequence $\langle \mathcal U_n:n\in\naturals\rangle $ of elements of $\mathcal O$ and each dense subset $D$ of $X$ there is a sequence $\langle K_n:n\in \naturals\rangle$ of elements of $\mathcal K$ such that each $A_n$ is a subset of $D$ and $\{{\rm St}(K_n,\mathcal U_n):n\in \naturals\} \in \mathcal B$. \end{definition} In this definition we have the following classes of spaces: \smallskip (1) selectively $(\mathcal O,\mathcal O)$-$(a)_{{\rm finite}}$-spaces are called \emph{absolutely strongly star-Menger spaces} (shortly {\sf ASSM} spaces), which form a subclass of ${\sf SSM}$-spaces; (2) selectively $(\mathcal O, \Gamma)$-$(a)_{{\rm finite}}$ spaces are \emph{absolutely strogly star-Hurewicz spaces} (shortly = {\sf ASSH} spaces), which form a subclass of {\sf SSH} spaces;, (3) ${\sf Sel}(\mathcal O,\mathcal O)$-$(a)_{{\rm singleton}}$ is the class of \emph{absolutely strongly Rothberger spaces} ({\sf ASSR} spaces for short), a subclass of the class {\sf SSR}; (4) For a space $X$ satisfying ${\sf Sel}(\mathcal O,\mathcal O)$-$(a)_{\rm closed \, discrete}$ we say that $X$ is a \emph{selectively $(a)$-space}, and this is a direct generalization of the notion of $(a)$-spaces. This class of spaces will be discussed in a separate subsection of this section. \medskip The following diagram shows relationships among the classes of spaces that we have defined so far. Let us mention that arrows in this diagram are not reversible; for some of them it was already demonstrated by examples in the previous sections, and for some other it will be done in what follows. \newpage \begin{center} \begin{picture}(260,260) \put(30,25){{\sf SR}} \put(55,28){\vector(1,0){35}} \put(107,25){{\sf SM}} \put(170,28){\vector(-1,0){35}} \put(180,25){{\sf SH}} \put(30,105){{\sf NSR}} \put(35,100){\vector(0,-1){55}} \put(55,108){\vector(1,0){35}} \put(107,105){{\sf NSM}} \put(112,100){\vector(0,-1){55}} \put(170,108){\vector(-1,0){35}} \put(180,105){{\sf NSH}} \put(185,100){\vector(0,-1){55}} \put(30,185){{\sf SSR}} \put(55,188){\vector(1,0){35}} \put(35,180){\vector(0,-1){55}} \put(107,185){{\sf SSM}} \put(170,188){\vector(-1,0){35}} \put(112,180){\vector(0,-1){55}} \put(180,185){{\sf SSH}} \put(185,180){\vector(0,-1){55}} \put(233,185){{\sf ASSH}} \put(225,188){\vector(-1,0){25}} %\put(233,125){{\sf SSC}} \put(238,120){\vector(0,-1){120}} %\put(233,-20){{\sf SC}}\put(228,145){\vector(-1,1){35}} %\put(230,-10){\vector(-1,1){30}} \put(30,245){{\sf R}} \put(35,240){\vector(0,-1){35}} \put(55,248){\vector(1,0){35}} \put(107,245){{\sf M}} \put(112,240){\vector(0,-1){35}} \put(170,248){\vector(-1,0){35}} \put(180,245){{\sf H}} \put(185,240){\vector(0,-1){35}} \put(-20,185){{\sf ASSR}} \put(10,188){\vector(1,0){15}} \put(-17,180){\vector(0,-1){40}} \put(-20,125){{\sf ASSM}} \put(0,140){\vector(2,1){85}} %\put(5,120){\vector(0,-1){60}} %\put(0,45){{\sf NSL}} \put(110,100) {\vector(-2,-1){85}} %\put(5,40){\vector(0,-1){57}} %\put(5,-30){{\sf SL}} \put(110,20){\vector(-2,-1){85}} \end{picture}\\ {\sf Diagram 1: Star selection properties} \end{center} %\newpage \subsection{{\sf ASSM, ASSH, ASSR} spaces} In this subsection we review very few basic results and examples concerning {\sf ASSM, ASSH, ASSR} spaces. We begin with some examples. \begin{example} \rm (1) The Tychonoff plank is a Tychonoff {\sf ASSM} which is not {\sf acc}. \smallskip (2) (\cite[Examples 2.1]{song-cejm}) There exists a Tychonoff {\sf ASSH} space $X$ which is not {\sf acc}. Let $X = [0,\omega]\times [0,\omega] \setminus \{(\omega,\omega\}$ as a subspace of the product $[0, \omega]\times [0,\omega]$. \smallskip (3) (\cite[Example 2.2]{song-cejm}) There exists a Tychonoff {\sf SSH} space $X$ which is not {\sf ASSH}. Such a space is $X= [0,\omega_1)\times[0,\omega_1]$. \end{example} Here are some properties of absolute star selection properties. First, similarly to other star selection properties, these properties are not hereditary. In \cite{song-assm} and \cite{song-cejm}, it is proved that in the class of Thychonoff spaces {\sf ASSM} and {\sf ASSH} properties are not preserved by regular-closed $G_{\delta}$-subspaces. \medskip Song noticed also that {\sf ASSM} and {\sf ASSH} properties are not invariants of continuous mappings. But he proved that these two properties, similarly to the {\sf acc} property \cite{matveev-acc}, are preserved by continuous varpseudoopen mappings. Recall that a continuous mapping $f: X \to Y$ is \emph{varpseudoopen} provided ${\rm int}_Y (f(U))\neq\emptyset$ for every nonempty open set $U$ of $X$. \smallskip Theorem 2.15 in \cite{song-cejm} states that if the the product of two spaces is {\sf ASSH}, then both spaces are {\sf ASSH}. On the other hand, in difference of some other star selection properties, the product of an {\sf ASSM} or {\sf ASSH} space $X$ and a compact space $Y$ need not be {\sf ASSM} or {\sf ASSH} as it was observed in \cite{song-assm} and \cite{song-cejm}. For both cases the product $[0,\omega_1) \times [0,\omega_1]$ can serve as an example. Matveev showed that the product of a Hausdorff {\sf acc} space and a first countable compact space is {\sf acc} (see \cite[Theorem 2.3]{matveev-acc}. So, it is naturally to ask: \begin{problem} {\rm (Song)} Is the product of an {\sf ASSH} space and a first countable compact space also {\sf ASSH} \end{problem} Let us finish with the following fact \cite[Theorem 3.8]{song-assm}: If $X$ is an {\sf ASSM} space with $e(X) < \omega_1$, then ${\sf AD}(X)$ is {\sf ASSM}. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Selectively $(a)$ and related spaces} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% The importance of property $(a)$ was established in the literature: there are strong connections of this property with countable compactness, normality and metrizability (see the already mentioned papers \cite{gauldvamana, justmatvszept, matveev97, rudinjerry, samuel-a, szeptvaughan}). Evidently, every $(a)$-space is selectively $(a)$. So, every monotonically normal space, in particular every $GO$-space, is selectively $(a)$, being an $(a)$-space (see \cite[Theorem 1]{rudinjerry}). For the same reason every selectively paracompact space is selectively $(a)$. It was observed in \cite{samuel} that every $T_1$, $\sigma$-compact space is selectively $(a)$. The Tychonoff plank is an example of a selectively $(a)$-space which is not an $(a)$-space (\cite[Example 2.6]{song-sel-a}). \smallskip Notice that every countably compact selectively $a$-space is ${\sf SSM}$, and every selectively $(\mathcal O,\Gamma)$-$(a)_{{\rm closed \, discrete}}$ space is {\sf SSH}. We will demonstrate similarities and differences between $(a)$-spaces and selectively $(a)$-spaces; in particular, we will show that there are many similarities between them. We begin with the following result which was stated in \cite{sel-(a)} without proof and which may be obtained by small changes in the proof of Lemma 1 and its corollary in \cite{matveev97}. \begin{theorem} \label{a-e} Let $X$ be a separable space. Then: \begin{itemize} \item[$(1)$] If $X$ is selectively $(a)$, then every closed discrete subset of $X$ has cardinality $< 2^{\omega}$; \item[$(2)$] If $X$ contains a discrete subspace having cardinality $\ge 2^{\omega}$, then $X^2$ is not hereditarily selectively $(a)$. \end{itemize} \end{theorem} In \cite{samuel}, the item (2) of this result was proved for a general case. \begin{theorem} {\rm (\cite[Theorem 3.1]{samuel})} If $X$ is a selectively $(a)$-space, then $X$ cannot contain closed and discrete subsets of size $\ge 2^{d(X)}$. \end{theorem} The following theorem is a nice strengthening of a result established in \cite{matveev97} by Matveev for $(a)$-spaces. \begin{theorem} {\rm (\cite[Theorem 3.4]{samuel})} Under ${\sf CH}$, separable, Moore, selectively $(a)$-spaces are metrizable. \end{theorem} It is shown in \cite[Theorem 3]{szeptvaughan} that there are $\Psi$-spaces which are $(a)$-spaces, hence selectively $(a)$, and those which are not $(a)$-spaces. It was observed in \cite{sel-(a)} that there are also $\Psi$-spaces which are not selectively $(a)$. For $\Psi$-spaces we have the following (Propositions 4.1 and 4.2, in \cite{samuel}). \begin{theorem} Let $\mathcal A$ be an almost disjoint family of subsets of $\mathbb N$. Then: \begin{itemize} \item[$(1)$] If $|\mathcal A| < \mathfrak d$, then $\Psi(\mathcal A)$ is selectively $(a)$; \item[$(2)$] If $\mathcal A$ is maximal, then $\Psi(\mathcal A)$ is selectively $(a)$ if and only if $|\mathcal A| < \mathfrak{d}$; \item[$(3)$] If $\mathfrak{p} = \mathfrak{c}$, then a $\Psi$-space satisfies property $(a)$ if and only if satisfies selectively $(a)$. \end{itemize} \end{theorem} (Here, $\mathfrak p$ is the pseudointersection number \cite{vaughan}.) It follows from this results that it is consistent that there are $\Psi$-spaces which are selectively $(a)$-spaces but not $(a)$-spaces. \begin{problem} $(1)$ {\rm (\cite[Question 5.3]{samuel})} Is it consistent that there is an almost disjoint family $\mathcal A$ of size $\mathfrak d$ such that $\Psi(\mathcal A)$ is selectively $(a)$ $(2)$ {\rm (\cite[Question 5.4]{samuel})} If $\Psi(\mathcal A)$ is normal, is it a selectively $(a)$-space $(3)$ {\rm (\cite[Question 5.5]{samuel})} If $\Psi(\mathcal A)$ is countably paracompact, is it a selectively $(a)$- space \end{problem} Let us notice that in \cite{song-sel-a} it was proved that assuming $2^{\aleph_0} = 2^{\aleph_1}$ there exists a normal space $X$ that is not selectively $(a).$ \medskip Generalizing a result of Szeptycki and Vaughan regarding characterization of property $(a)$ in $\Psi$-spaces, da Silva gave in \cite{samuel} the following combinatorial characterization of selectively $(a)$ $\Psi$-spaces. \begin{theorem} Let $\mathcal A = \{A_{\alpha}:\alpha <\kappa\} \subset \omega^{\omega}$ be an almost disjoint family of size $\kappa$. The corresponding space $\Psi(\mathcal A)$ is selectively $(a)$ if and only if the following property holds: for every sequence $\{f_n : n < \omega\}$ in $\omega^{\omega}$ there is a sequence $\{P_n : n <\omega\}$ of subsets of $\omega$ satisfying the following two conditions: (i) $|P_n\cap A_n| < \omega$ for all $n\in\omega$ and all $\alpha < \kappa$; (ii) for every $\alpha < \kappa$ there is $n\in\omega$ such that $P_n cap A_{\alpha} \nsubseteq f_n(\alpha)$. \end{theorem} In \cite{morgan-samuel} the authors proved that a certain effective parametrized weak diamond principle is enough to ensure countability of the almost disjoint family in this setting. In \cite[Corollary 3.3]{morgan-samuel} it was observed that selectively $(a)$-spaces from almost disjoint families are necessarily countable under some additional set-theoretic assumptions, and concluded that it follows that the statement ``all selectively $(a)$-spaces are countable" is consistent with ${\sf CH}$. These authors also noticed that there are no selectively $(a)$ almost disjoint families of size $\mathfrak{c}$; on the other hand, countable almost disjoint families are associated to metrizable $\Psi$-spaces, so if $\mathcal A$ is countable, then $\Psi(\mathcal A)$ is paracompact and therefore it is $(a)$ (thus, selectively $(a)$). \medskip The following results show the behaviour of selectively $(a)$-type spaces under mappings and basic operations with spaces. It is trivial that the selective $(a)$ property is not a hereditary property. It is also true in case of some special subspaces, for example, regular closed subspaces. \begin{theorem}\label{aimages} {\rm (\cite{sel-(a)})} A closed-and-open image $Y=f(X)$ of a selectively $(a)$-space $X$ is also selectively $(a)$. \end{theorem} The product of two selectively $(a)$-spaces need not be selectively $(a)$; the Sorgenfrey line $S$ and its square $S^2$ can serve as an example (by Theorem \ref{a-e} $S^2$ is not selectively $(a)$). It would be interesting to answer the following question posed in \cite{sel-(a)} (compare with \cite[Theorem 16]{justmatvszept}): \begin{problem} Is the product of a selectively $(a)$-space $X$ and a metrizable compact space $Y$ selectively $(a)$ \end{problem} We have the following \begin{theorem}\label{aproduct} {\rm (\cite{sel-(a)})} If the product $X\times Y$ of a space $X$ and a compact space $Y$ is selectively $(a)$, then $X$ is selectively $(\mathcal O,\mathcal O)$-$(a)_{{\rm closed}}$. \end{theorem} Now we consider when ${\sf AD}$ spaces have some of properties under consideration. \begin{theorem} \label{ax-sel-a} If $X \in {\sf Sel}(\mathcal O,\mathcal O)$-$(a)_{{\rm discrete}}$ and $e({\sf AD}(X)) < \omega_1$, then ${\sf AD}(X)$ is also in ${\sf Sel}(\mathcal O,\mathcal O)$-$(a)_{{\rm discrete}}$. \end{theorem} Another result of the same sort was proved in \cite{song-sel-a}: If $X$ is a normal selectively $(a)$-space with $e(X) < \omega_1$, then ${\sf AD}(X)$ is selectively $(a)$. Similarly, in \cite{sel-(a)} it was proved: \begin{theorem} \label{sel-a-ax} If the Alexandroff duplicate ${\sf AD}(X)$ of a space $X$ is selectively $(\mathcal O,\mathcal O)$-$(a)_{{\rm countable}}$, then $e(X) < \omega_1$. \end{theorem} In \cite[Question 2.11]{sel-(a)}, the authors asked if a space $X$ is selectively $(a)$ provided the space ${\sf AD}(X)$ is selectively $(a)$ This question was answered in \cite{song-sel-a}: there exists a Tychonoff countably compact space $X$ such that ${\sf AD}(X)$ is selectively $(a)$, but $X$ is not selectively $(a)$. We close this subsection by one more natural question of this kind: when subspaces of the Alexandroff duplicate ${\sf AD}(X)$ of a space $X$ have properties of selectively $(a)$-type. We have the following: \begin{theorem} {\rm (\cite{sel-(a)})} Let $A$ and $B$ be subspaces of a space $X$ such that $\overline{A} \cap B = \emptyset$ and $Z = (A\times \{1\}) \cup (B\times \{0\})$. If $e(Z) < \omega_1$ and $B$ is selectively $(\mathcal O,\mathcal O)$-$(a)_{{\rm discrete}}$, then $Z$ is selectively $(\mathcal O,\mathcal O)$-$(a)_{{\rm discrete}}$. \end{theorem} %%%%%%%%%%%%%%%%% 66666 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Uniform selection principles} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \bigskip In \cite{koc-unif} we have defined selection properties in uniform spaces and demonstrated that selection principles in uniform spaces are a good application of star selection principles to concrete special classes of spaces. The exposition in this section is based mainly on the paper \cite{koc-unif}, although the approach in this article is different from (but equivalent to) the approach in \cite{koc-unif}. \medskip Recall two equivalent approaches to the definition of uniform spaces; one is to define a uniformity on a set $X$ in terms of uniform covers, and the second to define it by using entourages of the diagonal \cite{engelking}. The first approach allows to define uniform selection principles similarly to definitions of the usual topological selection principles. By using this way we showed in \cite{koc-unif} that uniform selection principles are a kind of star selection properties as well as a kind of strongly star selection properties. Then we passed to description of uniform selection principles in terms of entourages of the diagonal. \medskip Recall some definitions and facts about uniform spaces. \smallskip A \emph{quasi-uniformity} on a set $X$ is a filter $\mathbb U$ on $X\times X$ satisfying the following two conditions: \begin{itemize} \item[$(QU1)$] $\Delta_X \subset U$ for each $U\in\mathbb U$; \item[$(QU2)$] For each $U\in\mathbb U$ there is $V\in\mathbb U$ such that $V\circ V \subset U$, \end{itemize} where $\Delta_X = \{(x,x):x\in X\}$ is the diagonal of $X$, and $V\circ V =\{(x,y)\in X\times X: \exists z\in X \mbox{ with } (x,z) \in V, (z,y)\in V\}$. The pair $(X,\mathbb U)$ is called a \emph{quasi-uniform space}. A quasi-uniformity $\mathbb U$ is a \emph{uniformity} on $X$, and $(X,\mathbb U)$ is a uniform space, if $\mathbb U$ satisfies also the condition \begin{itemize} \item[$(QU3)$] $U\in\mathbb U$ implies $U^{-1}\in\mathbb U$, \end{itemize} where $U^{-1} = \{(x,y)\in X\times X:(y,x)\in U\}$. For a subset $A$ of a (quasi-)uniform space $(X,\mathbb U)$ and $U\in\mathbb U$ we write \[ U[A]:= \{y\in X: (x,y)\in U \mbox{ for some } x\in A\}. \] We define uniform selection principles as follows. If $(X,\mathbb U)$ is a uniform space, then it is said to be: \begin{itemize} \item[{\sf UM:}] \emph{uniformly Menger} or \emph{{\sf M}-bounded} if for each sequence $\langle U_n:n\in\naturals\rangle$ of entourages of the diagonal of $X$ there is a sequence $\langle A_n:n\in\naturals \rangle $ of finite subsets of $X$ such that $X=\bigcup_{n\in\naturals}U_n[A_n]$. \item[{\sf U$\omega$M:}] \emph{$\omega$-{\sf M}-bounded} if for each sequence $\langle U_n:n\in\naturals\rangle $ of entourages of the diagonal of $X$ there is a sequence $\langle A_n:n\in\naturals\rangle$ of finite subsets of $X$ such that each finite subset of $X$ is contained in some $U_n[A_n]$. \item[{\sf UH:}] \emph{uniformly Hurewicz} or \emph{{\sf H}-bounded} if for each sequence $\langle U_n:n\in \naturals\rangle $ of elements of $\mathbb U$ there is a sequence $\langle A_n:n\in\naturals\rangle $ of finite subsets of $X$ such that each $x\in X$ belongs to all but finitely many sets $U_n[A_n]$. \item[{\sf UR:}] \emph{uniformly Rothberger} or \emph{{\sf R}-bounded} (resp. \emph{$\omega$-{\sf R}-bounded}) if for each sequence $\langle U_n:n\in\naturals\langle $ of entourages of the diagonal of $X$ there is a sequence $\langle x_n:n\in\naturals\langle$ of points in $X$ such that $X=\bigcup_{n\in\naturals}U_n[x_n]$ (resp. each finite subset of $X$ is contained in some $U_n[x_n]$. \end{itemize} \medskip \begin{remark} \rm It is evident that if a uniform space $X$ has the Menger property with respect to topology generated by the uniformity, then $X$ is {\sf M}-bounded. However, the converse need not be true: a non-Lindel\"of Tychonoff space is an example of {\sf M}-bounded space (with respect to the generated uniformity) which has no the Menger property. (Similar remarks hold for the {\sf R}-boundedness and {\sf H}-boundedness.) But a regular topological space $X$ has the Menger (Hurewicz, Rothberger) property if and only if its fine uniformity is {\sf M}-bounded ({\sf H}-bounded, {\sf R}-bounded). {\sf M}-bounded and especially {\sf H}-bounded uniform spaces have some properties which are similar to the corresponding properties of totally bounded uniform spaces. \end{remark} Recall that a uniform space $(X,\mathbb U)$ is said to be \emph{totally bounded} or \emph{precompact} (resp. \emph{pre-Lindel\"of} or \emph{$\omega$-bounded} if for each $U\in\mathbb U$ there is a finite (resp. countable) $A\subset X$ such that $U[A] = X$. It is understood that totally bounded uniform spaces are {\sf H}-bounded and thus {\sf M}-bounded and that {\sf M}-boundedness implies pre-Lindel\"ofness. \medskip The difference between uniform and topological selection principles is big enough \cite{koc-unif}. Here we point out some of differences on the example of Hurewicz properties. \medskip (1) Every subspace of an {\sf H}-bounded uniform space is {\sf H}-bounded. ({\sf M}- boundedness is also a hereditary property.) (2) A uniform space $X$ is {\sf H}-bounded if and only if its completion $\tilde{X}$ is {\sf H}-bounded. (3) The product of two {\sf H}-bounded uniform spaces is also {\sf H}-bounded. \medskip Let us mention that the product of two {\sf M}-bounded uniform spaces need not be {\sf M}-bounded (see the case of topological groups in the next subsection). \medskip We states the following two results from \cite{koc-unif} \begin{theorem} For a uniform space $(X,\mathbb U)$ the following are equivalent: \begin{itemize} \item[$(1)$] $X$ is $\omega$-{\sf M}-bounded; \item[$(2)$] For each sequence $\langle U_n:n\in\mathbb N\rangle$ of elements of $\mathbb U$ there is a sequence $\langle F_n:n\in\mathbb N\rangle$ of finite subsets of $X$ such that there is a sequence $n_1 < n_2 < \cdots$ such that each finite $A\subset X$ is contained in $\bigcup\{U_i[F_i]:n_k\le i