| Title:
|
Module-valued functors preserving the covering dimension (English) |
| Author:
|
Spěvák, Jan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
56 |
| Issue:
|
3 |
| Year:
|
2015 |
| Pages:
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377-399 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. \begin{enumerate} \item[(i)] If $G$ is a pathwise connected separable metric NSS abelian group and $X$, $Y$ are Tychonoff spaces such that the group-valued function spaces $C_p(X,G)$ and $C_p(Y,G)$ are topologically isomorphic as topological groups, then $\operatorname{dim} X=\operatorname{dim} Y$. \item[(ii)] If free precompact abelian groups of Tychonoff spaces $X$ and $Y$ are topologically isomorphic, then $\operatorname{dim} X=\operatorname{dim} Y$. \item[(iii)] If $R$ is a topological ring with a countable network and the free topological $R$-modules of Tychonoff spaces $X$ and $Y$ are topologically isomorphic, then $\operatorname{dim} X=\operatorname{dim} Y$. \end{enumerate} 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 $G$. (English) |
| Keyword:
|
covering dimension |
| Keyword:
|
topological group |
| Keyword:
|
function space |
| Keyword:
|
topology of pointwise convergence |
| Keyword:
|
free topological module |
| Keyword:
|
$l$-equivalence |
| Keyword:
|
$G$-equivalence |
| MSC:
|
54H11 |
| MSC:
|
54H13 |
| idZBL:
|
Zbl 06487001 |
| idMR:
|
MR3390284 |
| DOI:
|
10.14712/1213-7243.2015.131 |
| . |
| Date available:
|
2015-07-09T20:54:54Z |
| Last updated:
|
2017-10-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144352 |
| . |
| Reference:
|
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| Reference:
|
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| Reference:
|
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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| Reference:
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