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Subspaces of connected spaces. (English) Zbl 0855.54025
Summary: A connectification of a topological space $$X$$ is a connected Hausdorff space that contains $$X$$ as a dense subspace. S. W. Watson and R. G. Wilson [Houston J. Math. 19, 469-481 (1993; Zbl 0837.54012)] have noted that a Hausdorff space with a connectification has no nonempty proper clopen $$H$$-closed subspaces. Here it is proven that a Hausdorff space in which every nonempty proper clopen set is not feebly compact and the cardinality of the set of clopen sets is at most $$2^{\mathfrak c}$$ is connectifiable. This result is used to show that every metric space with no nonempty proper clopen $$H$$-closed subspace is connectifiable, answering a question asked by Watson and Wilson. Also, there is a nonconnectifiable, Hausdorff space of cardinality $${\mathfrak c}$$ with no proper $$H$$-closed subspace. Using the set-theoretic hypothesis $${\mathfrak p} = {\mathfrak c}$$, an example of a nonconnectifiable, normal Hausdorff space of cardinality $${\mathfrak c}$$ is constructed which has no nonempty compact open subset. This space is locally compact at all but one point, and if the continuum hypothesis is assumed it is first countable. This space provides a solution to questions asked by Watson and Wilson as well as Mack. The paper concludes by examining when extremally disconnected Tikhonov spaces have Tikhonov connectifications.

##### MSC:
 54D05 Connected and locally connected spaces (general aspects) 54D25 “$$P$$-minimal” and “$$P$$-closed” spaces 54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
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