Open connected sets in homogeneous spaces (Q2771473)

All topological spaces discussed here will be metric spaces. The results in this paper were obtained as a consequence of the author's investigation of the open problem as to whether every arc-connected homogeneous continuum is locally connected. Let \(U\subseteq X\), and let \(p\in X\). The union of all continua contained in \(U\) that contain \(p\) is denoted by \([p]_U\). Let \(A\subseteq B\subseteq X\). The collection of all points \(q\in A\) such that there exists a continuum \(C\subseteq X\) containing \(p\) and \(q\), and an open set \(U\) containing \(C\), such that \([p]_U\cap A\) is nowhere dense in \(B\) is denoted by \(M_p(A,B)\). The collection of all points \(q\in X\) such that for every continuum \(C\) containing both \(p\) and \(q\) and every open subset \(U\) containing \(C\), \((\overline {[p]_U})^\circ \neq\emptyset\) is denoted by \(E_p\) and for \(D\subseteq X\), \(E_D= \bigcap_{d\in D}E_d\). The paper contains a number of results dealing with these concepts. For example, the author proves that if \(X\) is separable, then \(M_p(A,B)\) is first category in \(B\) for each \(p\in X\). If \(X\) is also locally compact, then \(M_p(A,B)\) is a countable union of sets that are closed relative to \(A\) and nowhere dense relative to \(B\). A topological transformation group \((G,X)\) is referred to as a Polish transformation group if both \(G\) and \(X\) are Polish, that is, both are separable and completely metrizable. He shows that if \((G,X)\) is a transitive Polish transformation group and \(X\) is locally compact, then \(X\) is locally connected if and only if there is a countable dense subset \(D\subseteq X\) such that \(E_D\) has nonempty interior. He shows that a separable, homogeneous, arc-connected locally compact space \(X\) is locally connected if and only if there is a nonempty open subset \(U\) such that for each \(p\in U\) and each open subset \(V\) with \(p\in V\subseteq U\), \([p]_V\) is second category in \([p]_U\). Finally, he shows that a Polish, homogeneous, arc-connected space \(X\) is locally connected if and only if there is a nonempty open subset of \(X\) which has a component contained in a countable union of locally connected subspaces.