zbMATH — the first resource for mathematics

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.

54D05 Connected and locally connected spaces (general aspects)
54D25 “\(P\)-minimal” and “\(P\)-closed” spaces
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
Full Text: DOI
[1] Bell, M.G., On the combinatorial principle P(c), Fund. math., 114, 149-157, (1981) · Zbl 0581.03038
[2] Comfort, W.W.; Negrepontis, S., The theory of ultrafilters, (1974), Springer New York · Zbl 0298.02004
[3] van Douwen, E., Remote points, Dissertationes math., 188, 1-45, (1981) · Zbl 0525.54018
[4] van Douwen, E., The integers and topology, (), 111-167
[5] Emeryk, A.; Kulpa, W., The sorgenfrey line has no connected compactification, Comm. math. univ. carolin., 18, 483-487, (1977) · Zbl 0369.54007
[6] Engelking, R., General topology, () · Zbl 0684.54001
[7] Frolík, Z., A generalization of realcompact spaces, Czechoslovak math. J., 13, 127-137, (1963) · Zbl 0112.37603
[8] Gates, C.L., Some structural properties of the set of remote points of a metric space, Canad. J. math., 32, 195-209, (1980) · Zbl 0455.54019
[9] Gillman, L.; Jerison, M., Rings of continuous functions, (1960), Van Nostrand Reinhold New York · Zbl 0093.30001
[10] Hart, K.P., The čech-stone compactification of the real line, (), 317-352 · Zbl 0847.54028
[11] Isbell, J.R., Uniform spaces, () · Zbl 0124.15601
[12] Mack, J., The long line as a remainder, (), 237-245
[13] Magill, K.D., A note on compactifications, Math. Z., 94, 322-325, (1966) · Zbl 0146.18501
[14] van Mill, J.; Woods, R.G., Perfect images of zero-dimensional separable metric spaces, Canad. math. bull., 25, 41-47, (1982) · Zbl 0431.54006
[15] Plank, D., On a class of subalgebras of C(X) with applications to \(βX śb X\), Fund. math., 64, 41-45, (1969) · Zbl 0182.56302
[16] Porter, J.R.; Vermeer, J., Spaces with coarser minimal Hausdorff topologies, Trans. amer. math. soc., 289, 59-71, (1985) · Zbl 0577.54019
[17] Porter, J.R.; Woods, R.G., Extensions and absolutes of Hausdorff spaces, (1987), Springer New York
[18] Sierpinski, S., Sur une proprieté topologique des ensembles denumerable en soi, Fund. math., 1, 44-60, (1920)
[19] Tychonoff, A., Über die topologische erweiterung von Räumen, Math. ann., 102, 544-561, (1930) · JFM 55.0963.01
[20] Watson, S.W.; Wilson, R.G., Embeddings in connected spaces, Houston J. math., 19, 469-481, (1993) · Zbl 0837.54012
[21] Woods, R.G., Ideals of pseudocompact regular closed sets and absolutes of hewitt realcompactifications, Gen. topology appl., 2, 315-331, (1972) · Zbl 0267.54025
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.