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connected; H-closed; extensions; condensations
A non-connected, Hausdorff space with a countable network has a connected Hausdorff-subtopology iff the space is not-H-closed. This result answers two questions posed by Tkačenko, Tkachuk, Uspenskij, and Wilson [Comment. Math. Univ. Carolinae 37 (1996), 825--841]. A non-H-closed, Hausdorff space with countable $\pi $-weight and no connected, Hausdorff subtopology is provided.
[PT] Porter J.R., Tikoo M.L.: On Katětov spaces. Trans. Amer. Math. Soc. 289 (4) (1985), 59-71. MR 0779052
[PV] Porter J.R., Vermeer J.: Space with coarser minimal Hausdorff topologies. Canad. Math. Bull. 32.4 (1989), 425-433.
[PW1] Porter J.R., Woods R.G.: Extensions and Absolutes of Hausdorff Spaces. Springer-Verlag, Berlin, 1988. MR 0918341 | Zbl 0652.54016
[PW2] Porter J.R., Woods R.G.: Subspaces of connected spaces. Topology Appl. 68 (1996), 113-131. MR 1374077 | Zbl 0855.54025
[V] Vermeer J.: Private communication, 1984.
[TTUW] Tkačenko M.G., Tkachuk V.V., Uspenskij V.V., Wilson R.G.: In quest of weaker connected topologies. Comment. Math Univ. Carolinae 37.4 (1996), 825-841. MR 1440714
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