The Fr�chet-Urysohn compactum without points of countable character

1987 ◽  
Vol 41 (3) ◽  
pp. 210-216 ◽  
Author(s):  
V. I. Malykhin
Keyword(s):  
Countable Character

The Character of Certain Closed Sets

1984 ◽  
Vol 36 (1) ◽  
pp. 38-57 ◽  
Author(s):  
Mary Anne Swardson
Keyword(s):  
Topological Space ◽  
Closed Subset ◽  
Continuum Hypothesis ◽  
Closed Sets ◽  
Martin's Axiom ◽  
Martin’S Axiom ◽  
Countably Compact ◽  
Countable Character ◽  
Image Position ◽  
The Continuum

Let X be a topological space and let A ⊂ X. The character of A in X is the minimal cardinal of a base for the neighborhoods of A in X. Previous studies have shown that the character of certain subsets of X (or of X2) is related to compactness conditions on X. For example, in [12], Ginsburg proved that if the diagonalof a space X has countable character in X2, then X is metrizable and the set of nonisolated points of X is compact. In [2], Aull showed that if every closed subset of X has countable character, then the set of nonisolated points of X is countably compact. In [18], we noted that if every closed subset of X has countable character, then MA + ┐ CH (Martin's axiom with the negation of the continuum hypothesis) implies that X is paracompact.

A compact space having the cardinality of the continuum with no convergent sequences

1977 ◽  
Vol 81 (2) ◽  
pp. 177-181 ◽  
Author(s):  
V. V. Fedorčuk
Keyword(s):  
Compact Space ◽  
Convergent Sequence ◽  
Countable Subset ◽  
Countable Character ◽  
The Continuum ◽  
Convergent Sequences

1. Introduction. All spaces in this paper are Hausdorff. We recall that a space X is sequentially compact, if every countable subset of X contains a convergent sequence. Let us consider the three statements:(1) Every compact space of cardinality ≤ ʗ contains a point of countable character.(2) Every compact space of cardinality ≤ ʗ is sequentially compact.(3) Every infinite compact space of cardinality ≤ ʗ contains a convergent sequence.

scholarly journals Local compactness in free topological groups

2003 ◽  
Vol 68 (2) ◽  
pp. 243-265 ◽  
Author(s):  
Peter Nickolas ◽  
Mikhail Tkachenko
Keyword(s):  
Topological Group ◽  
Compact Set ◽  
Topological Groups ◽  
Locally Compact ◽  
Tychonoff Space ◽  
Abelian Topological Group ◽  
Local Compactness ◽  
Countable Type ◽  
Free Abelian Topological Group ◽  
Countable Character

We show that the subspace An(X) of the free Abelian topological group A(X) on a Tychonoff space X is locally compact for each n ∈ ω if and only if A2(X) is locally compact if an only if F2(X) is locally compact if and only if X is the topological sum of a compact space and a discrete space. It is also proved that the subspace Fn(X) of the free topological group F(X) is locally compact for each n ∈ ω if and only if F4(X) is locally compact if and only if Fn(X) has pointwise countable type for each n ∈ ω if and only if F4(X) has pointwise countable type if and only if X is either compact or discrete, thus refining a result by Pestov and Yamada. We further show that An(X) has pointwise countable type for each n ∈ ω if and only if A2(X) has pointwise countable type if and only if F2(X) has pointwise countable type if and only if there exists a compact set C of countable character in X such that the complement X \ C is discrete. Finally, we show that F2(X) is locally compact if and only if F3(X) is locally compact, and that F2(X) has pointwise countable type if and only if F3(X) has pointwise countable type.

Absolute C-Embedding of Spaces with Countable Character and Pseudocharacter Conditions

1980 ◽  
Vol 32 (4) ◽  
pp. 945-956 ◽  
Author(s):  
Alan Dow
Keyword(s):  
Closed Sets ◽  
Countably Compact ◽  
Neighborhood Base ◽  
Countable Compactness ◽  
Countable Character ◽  
Satisfying Property

Absolute C-embeddings have been studied extensively by C. E. Aull. We will use his notation P = C[Q] to mean that a space satisfying property Q is C-embedded in every space having property Q that it is embedded in if (and only if) it has property P. The first result of this type is due to Hewitt [5] where he proves that if Q is “Tychonoff” then P is almost compactness. Aull [2] proves that if Q is “T4 and countable pseudocharacter” or “T4 and first countable” then P is “countably compact”. In this paper we show that P is almost compactness if Q is “Tychonoff” and any of countable pseudocharacter, perfect, or first countability. Unfortunately for the last case we require the assumption that . Finally we show that P is countable compactness if Q is Tychonoff and “closed sets have a countable neighborhood base”. In each of the above results C-embedding may be replaced by C*-embeddings and the results hold if restricted to closed embeddings.

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

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
A. Dorantes-Aldama ◽  
R. Rojas-Hernández ◽  
Á. Tamariz-Mascarúa
Keyword(s):  
Ordered Set ◽  
Continuous Functions ◽  
Space Of Continuous Functions ◽  
Countable Space ◽  
Generalized Ordered Space ◽  
Ordered Space ◽  
Countable Character ◽  
Corson Compact ◽  
First Countable Space ◽  
Isolated Point

AbstractIn 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 Cp(X, Y) is the space of continuous functions from X to Y with the topology inherited from the Tychonoff product space YX. We write Cp(X) instead of Cp(X, R).We prove that for a first countable space Y, K(Cp(X, Y)) is not a lattice if any of the following cases happen:(a) Y is not locally compact,(b) X has only one non isolated point and Y is not compact.Furthermore, K(Cp(X)) is not a lattice when X satisfies one of the following properties:(i) X has a non-isolated point with countable character,(ii) X is not pseudocompact,(iii) X is infinite, pseudocompact and Cp(X) is normal,(iv) X is an infinite generalized ordered space.Moreover, K(Cp(X)) is not a lattice when X is an infinite Corson compact space, and for every space X, K(Cp(Cp(X))) is not a lattice. Finally, we list some unsolved problems.

scholarly journals On properties of countable character

1971 ◽  
Vol 4 (2) ◽  
pp. 183-192 ◽  
Author(s):  
Gabriel Sabbagh
Keyword(s):  
Direct Products ◽  
Similarity Type ◽  
Countable Character

It is proved that if a class X of algebras of countable similarity type is closed under isomorphism and ultrapower, then the class of subalgebras of direct products of elements of X is of countable character.

scholarly journals On cardinal invariants of space of finite subsets

2011 ◽  
Vol 52 ◽  
Author(s):  
Gintaras Praninskas
Keyword(s):  
Finite Subset ◽  
Cardinal Invariants ◽  
Countable Character ◽  
Network Weight

  In article some cardinal invariants of the space of finite subsets are examined. We obtained that such cardinal invariants us density and network weight, character and pi–character, pi–weight and weight coincide for any (A,B)–topology (Proposition 1). Also we obtained estimates of space of the finite subset of Lindelioft number and Suslin’s number (Theorem 2). Also is proved that for the space of finite subsets pseudo character is countable than and if than it’s diagonal number  is countable for any (A,B)–topology (Proposition 4). Characterization when space of finite subsets has countable character is given (Proposition 5).

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