What is a non-metrizable analog of metrizable compacta? I (Q409674)

This paper gives an answer to the question stated in the title. Metrizable compacta have the best properties from the point of view of general topology.NEWLINENEWLINE The author introduces two class of compacta, called metrcompacta and weak metrcompacta, which properly contain all metrizable compacta, and proves that several nice theorems holding for metrizable compacta remain true for such compacta.NEWLINENEWLINE A particularly interesting theorem is that every metrizable space has a compactification that is a metrcompactum. This shows the difference between metrizable compacta and metrcompacta, since non-separable metrizable spaces never have a metrizable compactification.NEWLINENEWLINE It also follows from this theorem that every metrizable space has a compactification that is a \(0\)-uniform Eberlein compactum, i.e., a compactum with a \(\sigma\)-disjoint \(T_0\)-separating family of open sets. This improves results by A. V. Arhangel'skii, T. Banakh and A. Leiderman.NEWLINENEWLINE On the other hand, metrcompacta and weak metrcompacta behave like metrizable compacta since they are both countable productive. Moreover, the following two applications are obtained. Factorization theorem:NEWLINENEWLINE For any continuous map \(f\) of a compactum \(X\) to a weak metrcompactnum \(Z\), there exist a weak metrcompactum \(Y\), a continuous onto map \(g:X\to Y\) and a continuous map \(h:Y\to Z\) such that \(f=h\circ g\), \(\dim Y\leq\dim X\) and \(w(Y)\leq w(Z)\).NEWLINENEWLINEThe coincidence of dimensions theorem:NEWLINENEWLINEFor any weak metrcompactum \(X\) and any integer \(n\in\{-1,0\}\cup \mathbb N\), the following properties (1)--(5) are equivalent: NEWLINENEWLINE(1) \(\dim X\leq n\); NEWLINENEWLINE(2) ind \(X\leq n\); NEWLINENEWLINE(3) Ind \(X\leq n\); NEWLINENEWLINE(4) \(\Delta X\leq n\), i.e., there exist a \(0\)-dimensional weak metrcompactum \(X_0\) and a continuous onto map \(f:X_0\to X\) such that \(|f^{-1}(x)|\leq n+1\) for each \(x\in X\); NEWLINENEWLINE(5) \(X\) is the union of Lindelöf \(p\)-spaces \(L_i\) with \(\dim L_i\leq 0\), \(i=0,1,\cdots,k\leq n+1\).NEWLINENEWLINEFinally, let us give the precise definitions of metrcompacta and weak metrcompacta. We need three definitions for a family \(\lambda\) of subsets of a space \(X\): NEWLINENEWLINE(i) \(\lambda\) is called \textit{jointly functionally open} if there is a continuous function \(f:X\to[0,1]\) such that \(\bigcup\lambda=f^{-1}(0,1]\) and for each \(U\in\lambda\), the function \(f_U\) defined by \(f_U(x)=f(x)\) for \(x\in U\) and \(f_U(x)=0\) for \(x\in X\setminus U\) is continuous. NEWLINENEWLINE(ii) \(\lambda\) is called \textit{weakly discrete in entourage \(O_\lambda\)} if \(\bigcup\lambda=O_\lambda\in\lambda\), all \(O\in\lambda\) is closed in \(O_\lambda\) and \(\lambda\setminus\{O_\lambda\}\) is disjoint. Moreover, \(\lambda\) is called \textit{dense} if, additionally, \(\bigcup(\lambda\setminus\{O_\lambda\})\) is dense in \(O_\lambda\). NEWLINENEWLINE(iii) \(\lambda\) is called \textit{\(T_0\)-separating} if for every pair of distinct points \(x,y\in X\), there is \(O\in\lambda\) with \(|O\cap\{x,y\}|=1\).NEWLINENEWLINE A compactum \(X\) is called a \textit{metrcompactum} if there exists a \(T_0\)-separating family \(\lambda=\bigcup_{i\in N}\lambda(i)\) of open sets in \(X\) such that each \(\lambda(i)\) is jointly functionally open and dense weakly discrete family in entourage \(O_{\lambda(i)}\). A compactum \(X\) is called a \textit{weak metrcompactum} if in the above definition, the \(\lambda(i)\) are not necessarily dense.