Local connectedness in transformation groups (Q2759043)

A function \(f\) is almost open if for each \(x\) in the domain of \(f\) and each open neighborhood \(U\) of \(x\), \(\overline {f(U)}\) is a neighborhood of \(f(x)\). Throughout the review, \((G,X)\) is a topological transformation group, \(p\) is a fixed element of \(X\) and \(T\) is the function from \(G\) to \(X\) which is defined by \(T(g)=gp\). In this paper, the author determines various conditions on \(G\) and \(T(G)\) which insure that \(T\) will be almost open. For example, for a cardinal number \(\kappa>0\), a subset \(A\) of a space \(Y\) is defined to be second \(\kappa\)-category in \(Y\) if \(A\) is not contained in the union of \(\leq\kappa\) nowhere dense subsets of \(Y\). The smallest cardinal number \(\kappa\) such that for each open neighborhood \(U\) of \(e\) in \(G\), there is a subset \(S'\) of \(S\) such that \(|S'|\leq \kappa\) and \(S\subseteq S'U\), is denoted by \(\mu(S)\). The author proves that the map \(T\) is almost open if and only if for each open neighborhood \(U\) of \(e\), there is a subset \(S'\) of \(G\) such that \(T(S'U)\) is second \(|S'|\)-category in \(X\). He also shows that \(T\) is almost open if \(G\) has a subset \(S\) such that \(T(S)\) is second \(\mu(S)\)-category in \(X\). The second portion of the paper is concerned with determining when \(T(G)\) and \(X\) are locally connected. The path component of \(e\) in \(G\) is denoted by \([e]_G\) and \(w(S)=\min \{|B|:B\) is a base for \(S\}+\omega\). The author shows that if \(X\) is regular and \(T([e]_G)\) is second \(w([e]_G)\)-category in \(X\), then \(X\) is connected im kleinen at each point of \(T(G)\) and \(T(G)\) is locally connected. One of the corollaries obtained is the fact that every topological group with a second category, second countable path component is locally connected. For any open neighborhood of \(x\in X\), \(C_U=T^{-1}((x)_U)\) where \((x)_U\) is the path component of \(x\) in \(U\). The author concludes by showing that if \(X\) is completely metrizable and \(G\) acts transitively on \(X\), then \(X\) is locally connected if and only if for each open neighborhood \(U\) of \(x\) there is a subset \(S'\subseteq G\) such that \(T(S'C_U)\) is second \(|S'|\)-category in \(X\).