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Relatively compact spaces and separation properties

Aleksander V. Arhangel'skiiIvan 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.

On AP and WAP spaces

Angelo BellaIvan V. Yashchenko — 1999

Commentationes Mathematicae Universitatis Carolinae

Several remarks on the properties of approximation by points (AP) and weak approximation by points (WAP) are presented. We look in particular at their behavior in product and at their relationships with radiality, pseudoradiality and related concepts. For instance, relevant facts are: (a) There is in ZFC a product of a countable WAP space with a convergent sequence which fails to be WAP. (b) C p over σ -compact space is AP. Therefore AP does not imply even pseudoradiality in function spaces, while...

On dense subspaces satisfying stronger separation axioms

We prove that it is independent of ZFC whether every Hausdorff countable space of weight less than c has a dense regular subspace. Examples are given of countable Hausdorff spaces of weight c which do not have dense Urysohn subspaces. We also construct an example of a countable Urysohn space, which has no dense completely Hausdorff subspace. On the other hand, we establish that every Hausdorff space of π -weight less than 𝔭 has a dense completely Hausdorff (and hence Urysohn) subspace. We show that...

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