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Keywords:
splittable; polyhedron; dimension
splittable; polyhedron; dimension
Summary:
A space $X$ is splittable over a space $Y$ (or splits over $Y$) if for every $A\subset X$ there exists a continuous map $f:X\rightarrow Y$ with $f^{-1} f A=A$. We prove that any $n$-dimensional polyhedron splits over $\bold R^{2n}$ but not necessarily over $\bold R^{2n-2}$. It is established that if a metrizable compact $X$ splits over $\bold R^n$, then $\dim X\leq n$. An example of $n$-dimensional compact space which does not split over $\bold R^{2n}$ is given.
References:
[1] Arhangel'skii A.V.: A general concept of splittability of a topological space (in Russian). in: Proceedings of the Fifth Tyraspol Symposium on General Topology and its Applications, Kishinev, Shtiintsa, 1985, 8-10.
[2] Arhangel'skii A.V., Shakhmatov D.B.: Splittable spaces and questions of functions approximations (in Russian). in: Proceedings of the Fifth Tyraspol Symposium on General Topology and its Applications, Kishinev, Shtiintsa, 1985, 10-11.
[3] Arhangel'skii A.V., Shakhmatov D.B.: On pointwise approximation of arbitrary functions by countable families of continuous functions (in Russian). Trudy Seminara I.G. Petrovskogo 13 (1988), 206-227. MR 0961436
[4] Tkachuk V.V.: Approximation of $\bold R^X$ with countable subsets of $C_p(X)$ and calibers of the space $C_p(X)$. Comment. Math. Univ. Carolinae 27 (1986), 267-276. MR 0857546
[5] Bregman Yu.H., Šapirovskii B.E., Šostak A.P.: On partition of topological spaces. Časopis pro pěstování mat. 109 (1984), 27-53. MR 0741207
[6] Mazurkiewicz S.: Sur les problèmes $\kappa $ et $\lambda $ de Urysohn. Fund. Math. 10 (1927), 311-319.
[7] Tkachuk V.: Remainders over discrete spaces - some applications (in Russian). Vestnik MGU, Mat., Mech., no. 4, 1990, 18-21. MR 1086601
[8] Malyhin V.I.: $\beta N$ is prime. Bull. Acad. Polon. Sci., Ser. Mat. 27 (1979), 295-297. MR 0552052 | Zbl 0433.54015
[9] Engelking R.: General Topology. PWN, Warszawa, 1977. MR 0500780 | Zbl 0684.54001
[10] Skljarenko E.G.: A theorem on maps, lowering dimension (in Russian). Bull. Acad. Polon. Sci., Ser. Mat. 10 (1962), 429-432. MR 0149445
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