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 -weight and no connected, Hausdorff subtopology is provided.
The cardinal functions of pseudocharacter, closed pseudocharacter, and character are used to examine H-closed spaces and to contrast the differences between H-closed and minimal Hausdorff spaces. An H-closed space is produced with the properties that and .
Si provano nuovi risultati riguardanti gli «-sets» e gli spazi «Near-compact». Si completano alcune ricerche pubblicate dai primi due autori nel 1978 e si risolvono due problemi recentemente posti da Cammaroto, Gutierrez, Nordo e Prada.
An -space is a topological space in which the topology is generated by the family of all -sets (see [N]). In this paper, minimal--spaces (where denotes several separation axioms) are investigated. Some new characterizations of -spaces are also obtained.
Maximal pseudocompact spaces (i.e. pseudocompact spaces possessing no strictly stronger pseudocompact topology) are characterized. It is shown that submaximal pseudocompact spaces whose pseudocompact subspaces are closed need not be maximal pseudocompact. Various techniques for constructing maximal pseudocompact spaces are described. Maximal pseudocompactness is compared to maximal feeble compactness.
The notion of a Hausdorff function is generalized to the concept of H-closed function and the concept of an H-closed extension of a Hausdorff function is developed. Each Hausdorff function is shown to have an H-closed extension.
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