The character of free topological groups II

Nickolas, Peter; Tkachenko, Mikhail

doi:10.4995/agt.2005.1962

Cited by 10 publications

(11 citation statements)

References 9 publications

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“…For example, the character of free abelian topological groups is only known in some cases (cf. [24,25]). The free abelian topological group A(X) over a Tychonoff space X is the abelian topological group with the universal property that each continuous function ϕ from X into any abelian topological group H has a unique extension to a continuous homomorphism φ : A(X) → H.…”

Section: Overview and Main Resultsmentioning

confidence: 99%

The character of topological groups, via bounded systems, Pontryagin–van Kampen duality and pcf theory

et al. 2014

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The Birkhoff-Kakutani Theorem asserts that a topological group is metrizable if and only if it has countable character. We develop and apply tools for the estimation of the character for a wide class of nonmetrizable topological groups.We consider abelian groups whose topology is determined by a countable cofinal family of compact sets. These are the closed subgroups of Pontryagin-van Kampen duals of metrizable abelian groups, or equivalently, complete abelian groups whose dual is metrizable. By investigating these connections, we show that also in these cases, the character can be estimated, and that it is determined by the weights of the compact subsets of the group, or of quotients of the group by compact subgroups. It follows, for example, that the density and the local density of an abelian metrizable group determine the character of its dual group. Our main result applies to the more general case of closed subgroups of Pontryagin-van Kampen duals of abelianČech-complete groups.In the special case of free abelian topological groups, our results extend a number of results of Nickolas and Tkachenko, which were proved using combinatorial methods.In order to obtain concrete estimations, we establish a natural bridge between the studied concepts and pcf theory, that allows the direct application of several major results from that theory. We include an introduction to these results and their use.

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Section: Overview and Main Resultsmentioning

confidence: 99%

The character of topological groups, via bounded systems, Pontryagin–van Kampen duality and pcf theory

et al. 2014

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“…Denote by d the cofinality of the partially ordered set N N . The next theorem gives a partial answer to the aforementioned question and provides an alternative and simple proof of the equality χ(A(X)) = χ(F (X)) = d for a non-discrete MK ω -space X (which is one of the principal results of [35]). Also this theorem generalizes Theorem 4.16 of [19] and gives an affirmative answer to Question 4.17 of [19].…”

Section: Introductionmentioning

confidence: 98%

On topological spaces and topological groups with certain local countable networks

Gabriyelyan

Ka̧kol

2015

Topology and its Applications

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The strong Pytkeev property Small base Baire space Topological group Function space (Free) locally convex space Free (abelian) topological groupBeing motivated by the study of the space C c (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 MK ω -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.

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“…In this paper and its sequel [14], we investigate these characters systematically and in some detail. Most of our results are in fact for free and free abelian topological groups on uniform spaces, since this gives maximum generality and allows the derivation at will of bounds for the characters of free and free abelian groups over topological spaces.…”

Section: Introductionmentioning

confidence: 99%

“…Our main results on the characters in the non-abelian case make use of a new description of a neighborhood base at the identity in the free topological group on an arbitrary uniform space (Theorem 3.6). While it is easy to see that the character of the free abelian topological group is always less than or equal to that of the corresponding free topological group (Lemma 3.1), the inequality is in general strict, and the two characters may indeed differ arbitrarily largely (see [14]). Using our new description of the topology of the free topological group, however, we show that the characters of the free abelian and the free (non-abelian) topological groups are equal whenever the underlying uniform space is ω-narrow (Theorem 3.15).…”

Section: Introductionmentioning

confidence: 99%

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The character of free topological groups I

Nickolas

Tkachenko

2005

Appl.Gen.Topol.

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Abstract.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.2000 AMS Classification: Primary 22A05, 54H11, 54A25; Secondary 54D30, 54D45.

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The character of free topological groups II

Cited by 10 publications

References 9 publications

The character of topological groups, via bounded systems, Pontryagin–van Kampen duality and pcf theory

The character of topological groups, via bounded systems, Pontryagin–van Kampen duality and pcf theory

On topological spaces and topological groups with certain local countable networks

The character of free topological groups I

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