Abstract: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 Tka\v cenko, 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.
Keywords: connected, H-closed, extensions, condensations
AMS Subject Classification: 54C10, 54D05, 54D35