Geodesic
Connected Hausdorff subtopologies
Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 1, pp. 195-201.

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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.
Classification : 54A10, 54C10, 54D05, 54D35
Keywords: connected; H-closed; extensions; condensations 
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Porter, Jack. Connected Hausdorff subtopologies. Commentationes Mathematicae Universitatis Carolinae, Tome 42 (2001) no. 1, pp. 195-201. http://geodesic.mathdoc.fr/item/CMUC_2001__42_1_a14/