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The minimum uniform compactification of a metric space

R. Grant Woods (1995)

Fundamenta Mathematicae

It is shown that associated with each metric space (X,d) there is a compactification u d X of X that can be characterized as the smallest compactification of X to which each bounded uniformly continuous real-valued continuous function with domain X can be extended. Other characterizations of u d X are presented, and a detailed study of the structure of u d X is undertaken. This culminates in a topological characterization of the outgrowth u d ℝ n ∖ ℝ n , where ( ℝ n , d ) is Euclidean n-space with its usual metric.

The Niemytzki plane is ϰ -metrizable

Wojciech Bielas, Andrzej Kucharski, Szymon Plewik (2021)

Mathematica Bohemica

We prove that the Niemytzki plane is ϰ -metrizable and we try to explain the differences between the concepts of a stratifiable space and a ϰ -metrizable space. Also, we give a characterisation of ϰ -metrizable spaces which is modelled on the version described by Chigogidze.

The notion of closedness in topological categories

Mehmet Baran (1993)

Commentationes Mathematicae Universitatis Carolinae

In [1], various generalizations of the separation properties, the notion of closed and strongly closed points and subobjects of an object in an arbitrary topological category are given. In this paper, the relationship between various generalized separation properties as well as relationship between our separation properties and the known ones ([4], [5], [7], [9], [10], [14], [16]) are determined. Furthermore, the relationships between the notion of closedness and strongly closedness are investigated...

The partially pre-ordered set of compactifications of Cp(X, Y)

A. Dorantes-Aldama, R. Rojas-Hernández, Á. Tamariz-Mascarúa (2015)

Topological Algebra and its Applications

In the set of compactifications of X we consider the partial pre-order defined by (W, h) ≤X (Z, g) if there is a continuous function f : Z ⇢ W, such that (f ∘ g)(x) = h(x) for every x ∈ X. Two elements (W, h) and (Z, g) of K(X) are equivalent, (W, h) ≡X (Z, g), if there is a homeomorphism h : W ! Z such that (f ∘ g)(x) = h(x) for every x ∈ X. We denote by K(X) the upper semilattice of classes of equivalence of compactifications of X defined by ≤X and ≡X. We analyze in this article K(Cp(X, Y)) where...

The product of two ordinals is hereditarily dually discrete

M.Á. Gaspar-Arreola, F. Hernández-Hernández (2012)

Commentationes Mathematicae Universitatis Carolinae

In Dually discrete spaces, Topology Appl. 155 (2008), 1420–1425, Alas et. al. proved that ordinals are hereditarily dually discrete and asked whether the product of two ordinals has the same property. In Products of certain dually discrete spaces, Topology Appl. 156 (2009), 2832–2837, Peng proved a number of partial results and left open the question of whether the product of two stationary subsets of ω 1 is dually discrete. We answer the first question affirmatively and as a consequence also give...

The rank of the diagonal and submetrizability

Aleksander V. Arhangel'skii, Raushan Z. Buzyakova (2006)

Commentationes Mathematicae Universitatis Carolinae

Several topological properties lying between the submetrizability and the G δ -diagonal property are studied. We are mostly interested in their relationship to each other and to the submetrizability. The first example of a Tychonoff space with a regular G δ -diagonal but without a zero-set diagonal is given. The same example shows that a Tychonoff separable space with a regular G δ -diagonal need not be submetrizable. We give a necessary and sufficient condition for submetrizability of a regular separable...

The Rothberger property on C p ( Ψ ( 𝒜 ) , 2 )

Daniel Bernal-Santos (2016)

Commentationes Mathematicae Universitatis Carolinae

A space X is said to have the Rothberger property (or simply X is Rothberger) if for every sequence 〈 𝒰 n : n ∈ ω 〉 of open covers of X , there exists U n ∈ 𝒰 n for each n ∈ ω such that X = ⋃ n ∈ ω U n . For any n ∈ ω , necessary and sufficient conditions are obtained for C p ( Ψ ( 𝒜 ) , 2 ) n to have the Rothberger property when 𝒜 is a Mrówka mad family and, assuming CH (the Continuum Hypothesis), we prove the existence of a maximal almost disjoint family 𝒜 for which the space C p ( Ψ ( 𝒜 ) , 2 ) n is Rothberger for all n ∈ ω .

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