On Wallman compactifications of \(T_{0}\)-spaces and related questions (Q2883535)

If \(X\) is a dense subset of \(Y\) then \(Y\) is called a \textit{compressed extension} of \(X\) if there exists an infinite cardinal \(\kappa\) such that \(X\cap P\neq\emptyset\) for any non-empty \(G_\kappa\)-set \(P\subset Y\) and every closed subset of \(Y\) is the intersection of a family of closed \(G_\kappa\)-sets.NEWLINENEWLINEFor a space \(X\), denote by \({\mathcal F}(X)\) the family of all closed subsets of \(X\). A family \({\mathcal L}\subset {\mathcal F}(X)\) is a \textit{\(WS\)-ring} if \(\mathcal L\) is closed under finite unions and finite intersections. Say that a \(WS\)-ring \(\mathcal L\) is a \textit{\(WF\)-ring } if for every \(F\in {\mathcal L}\) we have the equality \(X\setminus F= \bigcup\{H\in {\mathcal L}: H\cap F=\emptyset\}\). The family \({\mathcal F}(X)\) is a \(WF\)-ring if and only if \(X\) is a \(T_1\)-space.NEWLINENEWLINEA \textit{\(g\)-compactification} of a space \(X\) is a pair \((Y,f)\) where \(Y\) is a compact \(T_0\)-space, \(f:X\to Y\) is a continuous map, the set \(f(X)\) is dense in \(Y\) and the set \(\{y\}\) is closed for any \(y\in Y\setminus f(X)\). If \(f\) is an embedding then \(Y\) is called a \textit{compactification} of \(X\) and it is considered that \(X\subset Y\) and \(f(x)=x\) for any \(x\in X\).NEWLINENEWLINEGiven an \(WS\)-ring \(\mathcal L\) on a space \(X\) let \(u(x,{\mathcal L})=\{F\in{\mathcal L}: x\in F\}\) and denote by \(U({\mathcal L},X)\) the family of all ultrafilters \(\xi\subset {\mathcal L}\). Consider the set \(\omega_{\mathcal L}X =U({\mathcal L},X) \cup\{u(x,{\mathcal L}): x\in X\}\) and the map \(\omega_{\mathcal L}: X\to \omega_{\mathcal L} X\) defined by the equality \(\omega_{\mathcal L}(x)= u(x,{\mathcal L})\) for every \(x\in X\). The set \(\omega_{\mathcal L}X\) is considered with the topology generated by the closed base \(\{\{\xi\in \omega_{\mathcal L} X: H\in \xi\}: H\in {\mathcal L}\}\). Then \((\omega_{\mathcal L} X, \omega_{\mathcal L})\) is a \(g\)-compactification of the space \(X\) and \(\mathcal L\) is a \(WF\)-ring if and only if \(\omega_{\mathcal L} X\) is a \(T_1\)-space.NEWLINENEWLINEA \(g\)-compactification \((Y,f)\) of a space \(X\) is called a Wallman--Shanin \(g\)-compactification of \(X\) if \((X,f)=( \omega_{\mathcal L} X, \omega_{\mathcal L})\) for some \(WS\)-ring \({\mathcal L} \subset {\mathcal F}(X)\). The Wallman compactification \(\omega X =\omega_{{\mathcal F}(X)} X\) is always a Wallman--Shanin compactification of \(X\). The following statement is the main result of the paper.NEWLINENEWLINE\noindent \textbf{Theorem.} If \((Y,f)\) is a compressed \(g\)-compactification of a space \(X\), then \((Y,f)\) is a Wallman--Shanin \(g\)-compactification of \(X\).