Concerning σ-Connectedness of Baire Spaces

A well known theorem of Sierpiński states that every compact connected Hausdorff space is σ-connected. Hence, if X is locally compact and Hausdorff and X is locally connected at x, then x has a σ-connected neighborhood. However, local connectedness at x is not a necessary condition for x to have a σ-connected neighborhood, because the whole space may be σ-connected without being locally connected at x. One of the purposes of the present paper is then to investigate which points of a given locally compact Hausdorff space have σ-connected neighborhoods. We find also sufficient conditions for a connected, hereditarily Baire space to be σ-connected and prove the impossibility of expressing a connected, Čech-complete, rim compact space as a countable infinite union of mutually disjoint compact sets. Finally, we introduce the concept of D-connected space and relate it to σ-connectedness.