Module-valued functors preserving the covering dimension

Jan Spěvák

Commentationes Mathematicae Universitatis Carolinae (2015)

  • Volume: 56, Issue: 3, page 377-399
  • ISSN: 0010-2628

Abstract

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We prove a general theorem about preservation of the covering dimension dim by certain covariant functors that implies, among others, the following concrete results.

If G G is a pathwise connected separable metric NSS abelian group and X X , Y Y are Tychonoff spaces such that the group-valued function spaces C p ( X , G ) C_p(X,G) and C p ( Y , G ) C_p(Y,G) are topologically isomorphic as topological groups, then dim X = dim Y \operatorname{dim} X=\operatorname{dim} Y .

If free precompact abelian groups of Tychonoff spaces X X and Y Y are topologically isomorphic, then dim X = dim Y \operatorname{dim} X=\operatorname{dim} Y .

If R R is a topological ring with a countable network and the free topological R R -modules of Tychonoff spaces X X and Y Y are topologically isomorphic, then dim X = dim Y \operatorname{dim} X=\operatorname{dim} Y .

 The classical result of Pestov [The coincidence of the dimensions dim of l -equivalent spaces, Soviet Math. Dokl. 26 (1982), no. 2, 380–383] about preservation of the covering dimension by l -equivalence immediately follows from item (i) by taking the topological group of real numbers as G .

How to cite

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Spěvák, Jan. "Module-valued functors preserving the covering dimension." Commentationes Mathematicae Universitatis Carolinae 56.3 (2015): 377-399. <http://eudml.org/doc/271625>.

@article{Spěvák2015,
abstract = {We prove a general theorem about preservation of the covering dimension $\operatorname\{dim\}$ by certain covariant functors that implies, among others, the following concrete results. The classical result of Pestov [The coincidence of the dimensions dim of $l$-equivalent spaces, Soviet Math. Dokl. 26 (1982), no. 2, 380–383] about preservation of the covering dimension by $l$-equivalence immediately follows from item (i) by taking the topological group of real numbers as $G$.},
author = {Spěvák, Jan},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {covering dimension; topological group; function space; topology of pointwise convergence; free topological module; $l$-equivalence; $G$-equivalence},
language = {eng},
number = {3},
pages = {377-399},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Module-valued functors preserving the covering dimension},
url = {http://eudml.org/doc/271625},
volume = {56},
year = {2015},
}

TY - JOUR
AU - Spěvák, Jan
TI - Module-valued functors preserving the covering dimension
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2015
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 56
IS - 3
SP - 377
EP - 399
AB - We prove a general theorem about preservation of the covering dimension $\operatorname{dim}$ by certain covariant functors that implies, among others, the following concrete results. The classical result of Pestov [The coincidence of the dimensions dim of $l$-equivalent spaces, Soviet Math. Dokl. 26 (1982), no. 2, 380–383] about preservation of the covering dimension by $l$-equivalence immediately follows from item (i) by taking the topological group of real numbers as $G$.
LA - eng
KW - covering dimension; topological group; function space; topology of pointwise convergence; free topological module; $l$-equivalence; $G$-equivalence
UR - http://eudml.org/doc/271625
ER -

References

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