We compare several conditions sufficient for maximal resolvability of topological spaces. We prove that a space is maximally resolvable provided that for a dense set and for each the -character of at is not greater than the dispersion character of . On the other hand, we show that this implication is not reversible even in the class of card-homogeneous spaces.
We consider when one-to-one continuous mappings can improve normality-type and compactness-type properties of topological spaces. In particular, for any Tychonoff non-pseudocompact space there is a such that can be condensed onto a normal (-compact) space if and only if there is no measurable cardinal. For any Tychonoff space and any cardinal there is a Tychonoff space which preserves many properties of and such that any one-to-one continuous image of , , contains a closed copy...
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.
In this paper we introduce a connected topology T on the set ℕ of positive integers whose base consists of all arithmetic progressions connected in Golomb’s topology. It turns out that all arithmetic progressions which are connected in the topology T form a basis for Golomb’s topology. Further we examine connectedness of arithmetic progressions in the division topology T′ on ℕ which was defined by Rizza in 1993. Immediate consequences of these studies are results concerning local connectedness of...