Open subgroups of free abelian topological groups

1993 ◽  
Vol 114 (3) ◽  
pp. 439-442 ◽  
Author(s):  
Sidney A. Morris ◽  
Vladimir G. Pestov
Keyword(s):  
Topological Group ◽  
Sufficient Conditions ◽  
Open Subgroup ◽  
Regular Space ◽  
Topological Groups ◽  
Covering Dimension ◽  
Abelian Topological Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Free Abelian Topological Group

We prove that any open subgroup of the free abelian topological group on a completely regular space is a free abelian topological group. Moreover, the free topological bases of both groups have the same covering dimension. The prehistory of this result is as follows. The celebrated Nielsen–Schreier theorem states that every subgroup of a free group is free, and it is equally well known that every subgroup of a free abelian group is free abelian. The analogous result is not true for free (abelian) topological groups [1,5]. However, there exist certain sufficient conditions for a subgroup of a free topological group to be topologically free [2]; in particular, an open subgroup of a free topological group on a kω-space is topologically free. The corresponding question for free abelian topological groups asked 8 years ago by Morris [11] proved to be more difficult and remained open even within the realm of kω-spaces. In the present paper a comprehensive answer to this question is obtained.

scholarly journals Invariant metrics on free topological groups

1973 ◽  
Vol 9 (1) ◽  
pp. 83-88 ◽  
Author(s):  
Sidney A. Morris ◽  
H.B. Thompson
Keyword(s):  
Topological Group ◽  
Invariant Metrics ◽  
Regular Space ◽  
Discrete Topology ◽  
Group Topology ◽  
Topological Groups ◽  
Abstract Group ◽  
Completely Regular ◽  
Free Topological Group ◽  
Totally Disconnected

For a completely regular space X, G(X) denotes the free topological group on X in the sense of Graev. Graev proves the existence of G(X) by showing that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). It is natural to ask if the topology given by these two-sided invariant pseudo-metrics on G(X) is precisely the free topological group topology on G(X). If X has the discrete topology the answer is clearly in the affirmative. It is shown here that if X is not totally disconnected then the answer is always in the negative.

scholarly journals VARIETIES OF ABELIAN TOPOLOGICAL GROUPS AND SCATTERED SPACES

2008 ◽  
Vol 78 (3) ◽  
pp. 487-495 ◽  
Author(s):  
CAROLYN E. MCPHAIL ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Scattered Spaces

AbstractThe variety of topological groups generated by the class of all abelian kω-groups has been shown to equal the variety of topological groups generated by the free abelian topological group on [0, 1]. In this paper it is proved that the free abelian topological group on a compact Hausdorff space X generates the same variety if and only if X is not scattered.

scholarly journals A Schreier theorem for free topological groups

1975 ◽  
Vol 13 (1) ◽  
pp. 121-127 ◽  
Author(s):  
Peter Nickolas
Keyword(s):  
Topological Group ◽  
Closed Subgroup ◽  
Open Subgroup ◽  
Necessary Condition ◽  
Topological Groups ◽  
Free Topological Group

M.I.Graev has shown that subgroups of free topological groups need not be free. Brown and Hardy, however, have proved that any open subgroup of the free topological group on a kw-space is again a free topological group: indeed, this is true for any closed subgroup for which a Schreier transversal can be chosen continuously. This note provides a proof of this result more direct than that of Brown and Hardy. An example is also given to show that the condition stated in the theorem is not a necessary condition for freeness of a subgroup. Finally, a sharpened version of a particular case of the theorem is obtained, and is applied to the preceding example.

Varieties of topological groups

1969 ◽  
Vol 1 (2) ◽  
pp. 145-160 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Topological Space ◽  
Free Group ◽  
Quotient Group ◽  
Sufficient Conditions ◽  
Algebraic Theory ◽  
Topological Groups ◽  
Free Basis ◽  
Free Topological Group ◽  
Necessary And Sufficient

We introduce the concept of a variety of topological groups and of a free topological group F(X, ) of on a topological space X as generalizations of the analogous concepts in the theory of varieties of groups. Necessary and sufficient conditions for F(X, ) to exist are given and uniqueness is proved. We say the topological group FM,(X) is moderately free on X if its topology is maximal and it is algebraically free with X as a free basis. We show that FM(X) is a free topological group of the variety it generates and that if FM(X) is in then it is topologically isomorphic to a quotient group of F(X, ). It is also shown how well known results on free (free abelian) topological groups can be deduced. In the algebraic theory there are various equivalents of a free group of a variety. We examine the relationships between the topological analogues of these. In the appendix a result similar to the Stone-Čech compactification is proved.

Free subgroups of free abelian topological groups

1986 ◽  
Vol 100 (2) ◽  
pp. 347-353 ◽  
Author(s):  
E. Katz ◽  
S. A. Morris ◽  
P. Nickolas
Keyword(s):  
Topological Group ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Free Subgroups ◽  
General Conditions

In this paper we prove a theorem which gives general conditions under which the free abelian topological group F(Y) on a space Y can be embedded in the free abeian topological group F(X) on a space X.

scholarly journals Free Abelian topological groups and the Pontryagin-Van Kampen duality

1995 ◽  
Vol 52 (2) ◽  
pp. 297-311 ◽  
Author(s):  
Vladimir Pestov
Keyword(s):  
Topological Group ◽  
Dimensional Space ◽  
Topological Spaces ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group

We study the class of Tychonoff topological spaces such that the free Abelian topological group A(X) is reflexive (satisfies the Pontryagin-van Kampen duality). Every such X must be totally path-disconnected and (if it is pseudocompact) must have a trivial first cohomotopy group π1(X). If X is a strongly zero-dimensional space which is either metrisable or compact, then A(X) is reflexive.

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.

Separable Quotients of Free Topological Groups

2019 ◽  
Vol 63 (3) ◽  
pp. 610-623 ◽  
Author(s):  
Arkady Leiderman ◽  
Mikhail Tkachenko
Keyword(s):  
Topological Group ◽  
Point Group ◽  
Topological Groups ◽  
Topological Abelian Group ◽  
Abelian Topological Group ◽  
Compact Spaces ◽  
Separable Subgroup ◽  
Pseudocompact Spaces ◽  
Free Abelian Topological Group ◽  
The One

AbstractWe study the following problem: For which Tychonoff spaces $X$ do the free topological group $F(X)$ and the free abelian topological group $A(X)$ admit a quotient homomorphism onto a separable and nontrivial (i.e., not finitely generated) group? The existence of the required quotient homomorphisms is established for several important classes of spaces $X$, which include the class of pseudocompact spaces, the class of locally compact spaces, the class of $\unicode[STIX]{x1D70E}$-compact spaces, the class of connected locally connected spaces, and some others.We also show that there exists an infinite separable precompact topological abelian group $G$ such that every quotient of $G$ is either the one-point group or contains a dense non-separable subgroup and, hence, does not have a countable network.

scholarly journals Free abelian topological groups on countable CW-complexes

1990 ◽  
Vol 41 (3) ◽  
pp. 451-456 ◽  
Author(s):  
Eli Katz ◽  
Sidney A. Morris
Keyword(s):  
Topological Group ◽  
Positive Integer ◽  
Unit Ball ◽  
Closed Subgroup ◽  
Differentiable Manifold ◽  
Closed Unit Ball ◽  
Topological Groups ◽  
Abelian Topological Group ◽  
Cw Complex ◽  
Free Abelian Topological Group

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).

scholarly journals A remark on free topological groups with no small subgroups

1974 ◽  
Vol 18 (4) ◽  
pp. 482-484 ◽  
Author(s):  
H. B. Thompson
Keyword(s):  
Topological Group ◽  
Free Group ◽  
Regular Space ◽  
Group Topology ◽  
Topological Properties ◽  
Topological Groups ◽  
Abstract Group ◽  
Completely Regular ◽  
Metric Topology ◽  
Totally Disconnected

For a completely regular space X let G(X) be the Graev free topological group on X. While proving G(X) exists for completely regular spaces X, Graev showed that every pseudo-metric on X can be extended to a two-sided invariant pseudo-metric on the abstract group G(X). The free group topology on G(X) is usually strictly finer than this pseudo-metric topology. In particular this is the case when X is not totally disconnected (see Morris and Thompson [7]). It is of interest to know when G(X) has no small subgroups (see Morris [5]). Morris and Thompson [6] showed that this is the case if and only if X admits a continuous metric. The proof relied on properties of the free group topology and it is natural to ask if G(X) with its pseudo-metric topology has no small subgroups when and only when X admits a continuous metric. We show that this is the case. Topological properties of G(X) associated with the pseudo-metric topology have recently been studied by Joiner [3] and Abels [1].

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