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.
García-Máynez, A. Concerning σ-Connectedness of Baire Spaces. Canadian journal of mathematics, Tome 32 (1980) no. 6, pp. 1438-1447. doi: 10.4153/CJM-1980-113-2
@article{10_4153_CJM_1980_113_2,
author = {Garc{\'\i}a-M\'aynez, A.},
title = {Concerning {\ensuremath{\sigma}-Connectedness} of {Baire} {Spaces}},
journal = {Canadian journal of mathematics},
pages = {1438--1447},
year = {1980},
volume = {32},
number = {6},
doi = {10.4153/CJM-1980-113-2},
url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1980-113-2/}
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