Generalized Cantor manifolds and homogeneity (Q2901925)

The authors call a non-empty class \(\mathcal C\) of topological spaces \textit{admissible} if, for any \(X\in {\mathcal C}\), the class \(\mathcal C\) contains all topological copies of \(X\) and every \(F_\sigma\)-subset of \(X\). A space \(X\) has the property \(C\) if for every sequence \(\{{\mathcal U}_n:n\in \omega\}\) of open covers of \(X\), there exists a sequence \(\{{\mathcal V}_n: n\in\omega\}\) of open disjoint families in \(X\) such that each \({\mathcal V}_n\) refines \({\mathcal U}_n\) and \(\bigcup\{{\mathcal V}_n :n\in\omega\}\) is a cover of \(X\).NEWLINENEWLINEA space \(X\) is said to be \textit{a Mazurkiewicz manifold with respect to an admissible class \(\mathcal C\)} if for any closed disjoint sets \(X_0,X_1\subset X\) both having non-empty interior in \(X\) and every \(F_\sigma\)-subset \(F\subset X\) with \(F\in {\mathcal C}\), there exists a continuum \(K\subset X\setminus F\) such that \(K\cap X_i\neq\emptyset\) for \(i=0,1\). The space \(X\) is called to be \textit{a Cantor manifold with respect to an admissible class \(\mathcal C\)}, if \(X\) cannot be represented as a union of two proper closed sets whose intersection belongs to \(\mathcal C\).NEWLINENEWLINEThe authors prove several relative versions of Alexandroff's theorem about existence of an \(n\)-dimensional Cantor manifold in every \(n\)-dimensional compact space. One of the main results of the paper is the following fact:NEWLINENEWLINE\textbf{Theorem.} \textsl{Any compact space without the property \(C\) contains a closed set that is a Mazurkiewicz manifold with respect to the class of paracompact spaces with the property \(C\).}NEWLINENEWLINEThe authors also use some dimension-like invariants to define four classes of spaces; they prove that if \(\mathcal C\) is one of them and a metrizable continuum \(X\) does not belong to \(\mathcal C\) then \(X\) must be a Cantor manifold with respect to \(\mathcal C\).