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Relatively compact spaces and separation properties

Aleksander V. Arhangel'skii, Ivan V. Yashchenko (1996)

Commentationes Mathematicae Universitatis Carolinae

We consider the property of relative compactness of subspaces of Hausdorff spaces. Several examples of relatively compact spaces are given. We prove that the property of being a relatively compact subspace of a Hausdorff spaces is strictly stronger than being a regular space and strictly weaker than being a Tychonoff space.

Remarks on dense subspaces

Eva Murtinová (2004)

Czechoslovak Mathematical Journal

Some constructions of spaces with/without dense subspaces satisfying stronger separation axioms are presented.

Sequential compactness vs. countable compactness

Angelo Bella, Peter Nyikos (2010)

Colloquium Mathematicae

The general question of when a countably compact topological space is sequentially compact, or has a nontrivial convergent sequence, is studied from the viewpoint of basic cardinal invariants and small uncountable cardinals. It is shown that the small uncountable cardinal 𝔥 is both the least cardinality and the least net weight of a countably compact space that is not sequentially compact, and that it is also the least hereditary Lindelöf degree in most published models. Similar results, some definitive,...

Skeletally Dugundji spaces

Andrzej Kucharski, Szymon Plewik, Vesko Valov (2013)

Open Mathematics

We introduce and investigate the class of skeletally Dugundji spaces as a skeletal analogue of Dugundji space. Our main result states that the following conditions are equivalent for a given space X: (i) X is skeletally Dugundji; (ii) every compactification of X is co-absolute to a Dugundji space; (iii) every C*-embedding of the absolute p(X) in another space is strongly π-regular; (iv) X has a multiplicative lattice in the sense of Shchepin [Shchepin E.V., Topology of limit spaces with uncountable...

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