Two weak forms of countability axioms in free topological groups (original) (raw)

The character of free topological groups II

Applied General Topology, 2005

A systematic analysis is made of the character of the free and free abelian topological groups on metrizable spaces and compact spaces, and on certain other closely related spaces. In the first case, it is shown that the characters of the free and the free abelian topological groups on X are both equal to the "small cardinal" d if X is compact and metrizable, but also, more generally, if X is a non-discrete k!-space all of whose compact subsets are metrizable, or if X is a non-discrete Polish space. An example is given of a zero-dimensional separable metric space for which both characters are equal to the cardinal of the continuum. In the case of a compact space X, an explicit formula is derived for the character of the free topological group on X involving no cardinal invariant of X other than its weight; in particular the character is fully determined by the weight in the compact case. This paper is a sequel to a paper by the same authors in which the characters of the free groups were analysed under less restrictive topological assumptions.

Free topological groups over metrizable spaces

Topology and its Applications, 1989

Let X be a metrizable space and F(X) and A(X) be the free topological group over X and the free Abelian topological group over X respectively. We establish the following criteria:

The character of free topological groups I

Applied General Topology, 2005

A systematic analysis is made of the character of the free and free abelian topological groups on uniform spaces and on topological spaces. In the case of the free abelian topological group on a uniform space, expressions are given for the character in terms of simple cardinal invariants of the family of uniformly continuous pseudometrics of the given uniform space and of the uniformity itself. From these results, others follow on the basis of various topological assumptions. Amongst these: (i) if X is a compact Hausdorff space, then the character of the free abelian topological group on X lies between w(X) and w(X) ℵ 0 , where w(X) denotes the weight of X; (ii) if the Tychonoff space X is not a P-space, then the character of the free abelian topological group is bounded below by the "small cardinal" d; and (iii) if X is an infinite compact metrizable space, then the character is precisely d. In the non-abelian case, we show that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group, but the inequality is in general strict. It is also shown that the characters of the free abelian and the free topological groups are equal whenever the given uniform space is ω-narrow. A sequel to this paper analyses more closely the cases of the free and free abelian topological groups on compact Hausdorff spaces and metrizable spaces.

Free Topological Groups Over Ωµ-Metrizable Spaces

2015

Abstract. Let ωµ be an uncountable regular cardinal. For a Tychonoff space X, we let A(X) and F (X) be the free Abelian topological group and the free topological group over X, respectively. In this paper, we establish the next equivalences. Theorem. Let X be a space. The following are equivalent. 1. (X,UX) is an ωµ-metrizable uniform space, where UX is the universal uniformity on X. 2. A(X) is topologically orderable and χ(A(X)) = ωµ. 3. The derived set Xd is ωµ-compact and X is ωµ-metrizable. Theorem. Let X be a non-discrete space. Then, the following are equiva-lent. 1. X is ωµ-compact and ωµ-metrizable. 2. (X,UX) is ωµ-metrizable and X is ωµ-compact. 3. F (X) is topologically orderable and χ(F (X)) = ωµ. We also prove that an ωµ-metrizable uniform space (X,U) is a retract of its uniform free Abelian group A(X,U) and of its uniform free group F (X,U). 1.

Strong collective normality and countable compactness in free topological groups

Siberian Mathematical Journal, 1988

Assertion i. The Stone-Cech extension $(X n) of the space X n is naturally homeomorphic to the space (~X) n for each n~N+. 9 Assertion 2. Let F = F I • ... • F n be a product of closed sets in X and let 0 be an open neighborhood of the set F in X n. Then there exist open sets in X, OlaF1, ..., O~F~ such that Ol• 9

Free abelian topological groups on countable CW-complexes

Bulletin of the Australian Mathematical Society, 1990

Let n be a positive integer, Bn the closed unit ball in Euclidean n-space, and X any countable CW-complex of dimension at most n. It is shown that the free Abelian topological group on Bn, F(Bn), has F(X) as a closed subgroup. It is also shown that for every differentiable manifold Y of dimension at most n, F(Y) is a closed subgroup of F(Bn).

The topological structure of (homogeneous) spaces and groups with countable cs*-character

Applied General Topology

In this paper we introduce and study three new cardinal topological invariants called the cs*, cs-, and sb-characters. The class of topological spaces with countable cs*-character is closed under many topological operations and contains all aleph-spaces and all spaces with point-countable cs*-network. Our principal result states that each non-metrizable sequential topological group with countable cs*-character has countable pseudo-character and contains an open komegak_\omegakomega-subgroup.

Local compactness in free topological groups

Bulletin of the Australian Mathematical Society, 2003

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 ...

On topological spaces and topological groups with certain local countable networks

Topology and its Applications, 2015

Being motivated by the study of the space Cc(X) of all continuous real-valued functions on a Tychonoff space X with the compact-open topology, we introduced in [16] the concepts of a cp-network and a cn-network (at a point x) in X. In the present paper we describe the topology of X admitting a countable cp-or cn-network at a point x ∈ X. This description applies to provide new results about the strong Pytkeev property, already well recognized and applicable concept originally introduced by Tsaban and Zdomskyy [44]. We show that a Baire topological group G is metrizable if and only if G has the strong Pytkeev property. We prove also that a topological group G has a countable cp-network if and only if G is separable and has a countable cp-network at the unit. As an application we show, among the others, that the space D ′ (Ω) of distributions over open Ω ⊆ R n has a countable cp-network, which essentially improves the well known fact stating that D ′ (Ω) has countable tightness. We show that, if X is an MKω-space, then the free topological group F (X) and the free locally convex space L(X) have a countable cp-network. We prove that a topological vector space E is p-normed (for some 0 < p ≤ 1) if and only if E is Fréchet-Urysohn and admits a fundamental sequence of bounded sets.