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Strong remote points

Sergei Logunov (2002)

Commentationes Mathematicae Universitatis Carolinae

Remote points constructed so far are actually strong remote. But we construct remote points of another type.

The Baire property in remainders of topological groups and other results

Aleksander V. Arhangel'skii (2009)

Commentationes Mathematicae Universitatis Carolinae

It is established that a remainder of a non-locally compact topological group G has the Baire property if and only if the space G is not Čech-complete. We also show that if G is a non-locally compact topological group of countable tightness, then either G is submetrizable, or G is the Čech-Stone remainder of an arbitrary remainder Y of G . It follows that if G and H are non-submetrizable topological groups of countable tightness such that some remainders of G and H are homeomorphic, then the spaces...

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 subspace of weak P -points of ℕ *

Salvador García-Ferreira, Y. F. Ortiz-Castillo (2015)

Commentationes Mathematicae Universitatis Carolinae

Let W be the subspace of ℕ * consisting of all weak P -points. It is not hard to see that W is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that W is a p -pseudocompact space for all p ∈ ℕ * .

Two types of remainders of topological groups

Aleksander V. Arhangel'skii (2008)

Commentationes Mathematicae Universitatis Carolinae

We prove a Dichotomy Theorem: for each Hausdorff compactification b G of an arbitrary topological group G , the remainder b G ∖ G is either pseudocompact or Lindelöf. It follows that if a remainder of a topological group is paracompact or Dieudonne complete, then the remainder is Lindelöf, and the group is a paracompact p -space. This answers a question in A.V. Arhangel’skii, Some connections between properties of topological groups and of their remainders, Moscow Univ. Math. Bull. 54:3 (1999), 1–6. It is...

Weak extent in normal spaces

Ronnie Levy, Mikhail Matveev (2005)

Commentationes Mathematicae Universitatis Carolinae

If X is a space, then the weak extent we ( X ) of X is the cardinal min { α : If 𝒰 is an open cover of X , then there exists A ⊆ X such that | A | = α and St ( A , 𝒰 ) = X } . In this note, we show that if X is a normal space such that | X | = 𝔠 and we ( X ) = ω , then X does not have a closed discrete subset of cardinality 𝔠 . We show that this result cannot be strengthened in ZFC to get that the extent of X is smaller than 𝔠 , even if the condition that we ( X ) = ω is replaced by the stronger condition that X is separable.

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