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# Article

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Keywords:
covering dimension; topological group; function space; topology of pointwise convergence; free topological module; $l$-equivalence; $G$-equivalence
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 $G$.
References:
[1] Arhangel'skiĭ A.V.: The principle of $\tau$ approximation and a test for equality of dimension of compact Hausdorff spaces. (Russian), Dokl. Akad. Nauk SSSR 252 (1980), no. 4, 777-780. MR 0580830
[2] Dikranjan D., Shakhmatov D., Spěvák J.: NSS and TAP properties in topological groups close to being compact. ArXiv prerpint arXiv:0909.2381v1 [math.GN].
[3] Freyd P.: Abelian Categories. An Introduction to the Theory of Functors. Harper's Series in Modern Mathematics, Harper & Row, New York, 1964, 11+164 pp. MR 0166240 | Zbl 0121.02103
[4] Gul'ko S.P.: On uniform homeomorphisms of spaces of continuous functions. (Russian), Trudy Mat. Inst. Steklov. 193 (1992), 82–88; English translation: Proc. Steklov Inst. Math. 193 (1993), 87–93. MR 1265990 | Zbl 0804.54018
[5] Joiner Ch.: Free topological groups and dimension. Trans. Amer. Math. Soc. 220 (1976), 401–418. DOI 10.1090/S0002-9947-1976-0412322-X | MR 0412322 | Zbl 0331.54026
[6] Kelley J.L.: Obshchaya Topologiya. (Russian), [General topology], translated from English by A.V. Arhangel'skiĭ, edited by P.S. Aleksandrov, Izdat. “Nauka”, Moscow, 1968. MR 0239550 | Zbl 0518.54001
[7] Krupski M.: Topological dimension of a space is determined by the pointwise topology of its function space. ArXiv prerpint arXiv:1411.1549v1 [math.GN].
[8] Menini C., Orsatti A.: Dualities between categories of topological modules. Comm. Algebra 11 (1983), no. 1, 21–66. DOI 10.1080/00927878308822836 | MR 0687405 | Zbl 0507.16028
[9] Pavlovskiĭ D.S.: Spaces of continuous functions. (Russian), Dokl. Akad. Nauk SSSR 253 (1980), no. 1, 38–41. MR 0577006
[10] Pestov V.G.: The coincidence of the dimensions dim of $l$-equivalent spaces. Soviet Math. Dokl. 26 (1982), no. 2, 380–383.
[11] Shakhmatov D.B.: Baire isomorphisms at the first level and dimension. Topology Appl. 107 (2000), 153–159. MR 1783842 | Zbl 0960.54024
[12] Shakhmatov D.B., Spěvák J.: Group-valued continuous functions with the topology of pointwise convergence. Topology Appl. 157 (2010), 1518–1540. DOI 10.1016/j.topol.2009.06.022 | MR 2610463 | Zbl 1195.54040
[13] Spěvák J.: Finite-valued mappings preserving dimension. Houston J. Math. 31 (2011), no. 1, 327–348. MR 2786558 | Zbl 1213.54049
[14] Tkachuk V.V.: Duality with respect to the functor $C_p$ and cardinal invariants of the type of the Souslin number. (Russian), Mat. Zametki 37 (1985), no. 3, 441–451. MR 0790433
[15] Zambahidze L.G.: On relations between dimensional and cardinal functions of spaces imbedded in spaces of a special type. (Russian), Soobshch. Akad. Nauk Gruzin. SSR 100 (1980), no. 3, 557–560. MR 0615246
[16] Zambahidze L.G.: Relations between dimensions of free bases of free topological groups. (Russian), Soobshch. Akad. Nauk Gruzin. SSR 97 (1980), no. 3, 569–572. MR 0581033

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