Covering Dimension and Universality Property on Frames

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.

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Covering dimension and characteristic classes for foliations

Algebraic and Geometric Topology - Proceedings of Symposia in Pure Mathematics ◽

10.1090/pspum/032.2/520535 ◽

1978 ◽

pp. 189-190

Author(s):

Herbert Shulman

Keyword(s):

Characteristic Classes ◽

Covering Dimension

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Four-Dimension Equivalences

Canadian Journal of Mathematics ◽

10.4153/cjm-1968-006-7 ◽

1968 ◽

Vol 20 ◽

pp. 48-50 ◽

Cited By ~ 2

Author(s):

J. R. Gard ◽

R. D. Johnson

Keyword(s):

Topological Spaces ◽

Covering Dimension ◽

Closed Set ◽

Set Separation

The object of this paper is to establish the equivalence of four functionrelated dimension concepts in arbitrary topological spaces. These concepts involve stability of functions (3, p. 74), the modification of covering dimension involving basic covers (1, p. 243) (which is equivalent to Yu. M. Smirnov's definition using normal covers), the definition involving essential mappings (2, p. 496), and a modification of the closed set separation characterization of dimension in (3, p. 35).

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Covering dimension and finite spaces

Applied Mathematics and Computation ◽

10.1016/j.amc.2011.08.040 ◽

2011 ◽

Vol 218 (7) ◽

pp. 3122-3130 ◽

Cited By ~ 8

Author(s):

D.N. Georgiou ◽

A.C. Megaritis

Keyword(s):

Covering Dimension ◽

Finite Spaces

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