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.

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.

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).

EMBEDDING OF THE FREE ABELIAN TOPOLOGICAL GROUP INTO

Mathematika ◽  
2019 ◽  
Vol 65 (3) ◽  
pp. 708-718 ◽  
Author(s):  
Mikołaj Krupski ◽  
Arkady Leiderman ◽  
Sidney Morris
Keyword(s):  
Topological Group ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group

SUBSPACES OF THE FREE TOPOLOGICAL VECTOR SPACE ON THE UNIT INTERVAL

2017 ◽  
Vol 97 (1) ◽  
pp. 110-118 ◽  
Author(s):  
SAAK S. GABRIYELYAN ◽  
SIDNEY A. MORRIS
Keyword(s):  
Topological Group ◽  
Vector Space ◽  
Topological Vector Space ◽  
Unit Interval ◽  
Vector Subspace ◽  
Locally Compact ◽  
Tychonoff Space ◽  
Abelian Topological Group ◽  
Free Abelian Topological Group ◽  
Usual Topology

For a Tychonoff space $X$, let $\mathbb{V}(X)$ be the free topological vector space over $X$, $A(X)$ the free abelian topological group over $X$ and $\mathbb{I}$ the unit interval with its usual topology. It is proved here that if $X$ is a subspace of $\mathbb{I}$, then the following are equivalent: $\mathbb{V}(X)$ can be embedded in $\mathbb{V}(\mathbb{I})$ as a topological vector subspace; $A(X)$ can be embedded in $A(\mathbb{I})$ as a topological subgroup; $X$ is locally compact.

scholarly journals Varieties of topological groups II

1970 ◽  
Vol 2 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Sidney A. Morris
Keyword(s):  
Free Group ◽  
Compact Hausdorff Space ◽  
Closed Subset ◽  
Hausdorff Space ◽  
Projective Group ◽  
Necessary And Sufficient Condition ◽  
Topological Groups ◽  
Finitely Generated ◽  
Tychonoff Space ◽  
Necessary And Sufficient

This paper is a sequel to one entitled “Varieties of topological groups”. The variety V of topological groups is said to be full if it contains every group which is algebraically isomorphic to a group in V For any Tychonoff space X, the free group F of V on X exists, is Hausdorff and disconnected, and has X as a closed subset. Any subgroup of F which is algebraically fully invariant is a closed subset of F. If X is a compact Hausdorff space, then F is normal. Let V be a full Schreier variety and X a Tychonoff space, then all finitely generated subgroups of F are free in V.A β-variety V is one for which the free group of V on each compact Hausdorff space exists and is Hausdorff. For any β-variety V and Tychonoff space X, the free group of V exists, is Hausdorff and has X as a closed subset. A necessary and sufficient condition for V to be a β-variety is given.The concept of a projective (topological) group of a variety V is introduced. The projective groups of V are shown to be precisely the summands of the free groups of V. A finitely generated Hausdorff projective group of a Schreier variety V is free in V.

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