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Relative normality and product spaces

Takao Hoshina, Ryoken Sokei (2003)

Commentationes Mathematicae Universitatis Carolinae

Arhangel’skiĭ defines in [Topology Appl. 70 (1996), 87–99], as one of various notions on relative topological properties, strong normality of A in X for a subspace A of a topological space X , and shows that this is equivalent to normality of X A , where X A denotes the space obtained from X by making each point of X ∖ A isolated. In this paper we investigate for a space X , its subspace A and a space Y the normality of the product X A × Y in connection with the normality of ( X × Y ) ( A × Y ) . The cases for paracompactness, more...

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.

Relatively realcompact sets and nearly pseudocompact spaces

John J. Schommer (1993)

Commentationes Mathematicae Universitatis Carolinae

A space is said to be nearly pseudocompact iff v X - X is dense in β X - X . In this paper relatively realcompact sets are defined, and it is shown that a space is nearly pseudocompact iff every relatively realcompact open set is relatively compact. Other equivalences of nearly pseudocompactness are obtained and compared to some results of Blair and van Douwen.

Remainders of metrizable and close to metrizable spaces

A. V. Arhangel'skii (2013)

Fundamenta Mathematicae

We continue the study of remainders of metrizable spaces, expanding and applying results obtained in [Fund. Math. 215 (2011)]. Some new facts are established. In particular, the closure of any countable subset in the remainder of a metrizable space is a Lindelöf p-space. Hence, if a remainder of a metrizable space is separable, then this remainder is a Lindelöf p-space. If the density of a remainder Y of a metrizable space does not exceed 2 ω , then Y is a Lindelöf Σ-space. We also show that many of...

Remarks on absolutely star countable spaces

Yan-Kui Song (2013)

Open Mathematics

We prove the following statements: (1) every Tychonoff linked-Lindelöf (centered-Lindelöf, star countable) space can be represented as a closed subspace in a Tychonoff pseudocompact absolutely star countable space; (2) every Hausdorff (regular, Tychonoff) linked-Lindelöf 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.

Remarks on sequence-covering maps

Luong Quoc Tuyen (2012)

Commentationes Mathematicae Universitatis Carolinae

In this paper, we prove that each sequence-covering and boundary-compact map on g -metrizable spaces is 1-sequence-covering. Then, we give some relationships between sequence-covering maps and 1-sequence-covering maps or weak-open maps, and give an affirmative answer to the problem posed by F.C. Lin and S. Lin in [Lin.F.C.and.Lin.S-2011].

Remarks on star countable discrete closed spaces

Yan-Kui Song (2013)

Czechoslovak Mathematical Journal

In this paper, we prove the following statements: (1) There exists a Tychonoff star countable discrete closed, pseudocompact space having a regular-closed subspace which is not star countable. (2) Every separable space can be embedded into an absolutely star countable discrete closed space as a closed subspace. (3) Assuming 2 ℵ 0 = 2 ℵ 1 , there exists a normal absolutely star countable discrete closed space having a regular-closed subspace which is not star countable.

Remarks on star covering properties in pseudocompact spaces

Yan-Kui Song (2013)

Mathematica Bohemica

Let P be a topological property. A space X is said to be star P if whenever 𝒰 is an open cover of X , there exists a subspace A ⊆ X with property P such that X = St ( A , 𝒰 ) , where St ( A , 𝒰 ) = ⋃ { U ∈ 𝒰 : U ∩ A ≠ ∅ } . In this paper, we study the relationships of star P properties for P ∈ { Lindel ö f , compact , countablycompact } in pseudocompact spaces by giving some examples.

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