A note on splittable spaces

Vladimir Vladimirovich Tkachuk

Commentationes Mathematicae Universitatis Carolinae (1992)

  • Volume: 33, Issue: 3, page 551-555
  • ISSN: 0010-2628

Abstract

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A space X is splittable over a space Y (or splits over Y ) if for every A ⊂ X there exists a continuous map f : X → Y with f - 1 f A = A . We prove that any n -dimensional polyhedron splits over 𝐑 2 n but not necessarily over 𝐑 2 n - 2 . It is established that if a metrizable compact X splits over 𝐑 n , then dim X ≤ n . An example of n -dimensional compact space which does not split over 𝐑 2 n is given.

How to cite

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Tkachuk, Vladimir Vladimirovich. "A note on splittable spaces." Commentationes Mathematicae Universitatis Carolinae 33.3 (1992): 551-555. <http://eudml.org/doc/247369>.

@article{Tkachuk1992,
abstract = {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 $\mathbf \{R\}^\{2n\}$ but not necessarily over $\mathbf \{R\}^\{2n-2\}$. It is established that if a metrizable compact $X$ splits over $\mathbf \{R\}^n$, then $\dim X\le n$. An example of $n$-dimensional compact space which does not split over $\mathbf \{R\}^\{2n\}$ is given.},
author = {Tkachuk, Vladimir Vladimirovich},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {splittable; polyhedron; dimension; splittability; polyhedron},
language = {eng},
number = {3},
pages = {551-555},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {A note on splittable spaces},
url = {http://eudml.org/doc/247369},
volume = {33},
year = {1992},
}

TY - JOUR
AU - Tkachuk, Vladimir Vladimirovich
TI - A note on splittable spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1992
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 33
IS - 3
SP - 551
EP - 555
AB - 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 $\mathbf {R}^{2n}$ but not necessarily over $\mathbf {R}^{2n-2}$. It is established that if a metrizable compact $X$ splits over $\mathbf {R}^n$, then $\dim X\le n$. An example of $n$-dimensional compact space which does not split over $\mathbf {R}^{2n}$ is given.
LA - eng
KW - splittable; polyhedron; dimension; splittability; polyhedron
UR - http://eudml.org/doc/247369
ER -

References

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  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. (1988) MR0961436
  4. Tkachuk V.V., Approximation of 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. (1986) MR0857546
  5. Bregman Yu.H., Šapirovskii B.E., Šostak A.P., On partition of topological spaces, Časopis pro pěstování mat. 109 (1984), 27-53. (1984) MR0741207
  6. Mazurkiewicz S., Sur les problèmes κ et λ de Urysohn, Fund. Math. 10 (1927), 311-319. (1927) 
  7. Tkachuk V., Remainders over discrete spaces - some applications (in Russian), Vestnik MGU, Mat., Mech., no. 4, 1990, 18-21. MR1086601
  8. Malyhin V.I., β N is prime, Bull. Acad. Polon. Sci., Ser. Mat. 27 (1979), 295-297. (1979) Zbl0433.54015MR0552052
  9. Engelking R., General Topology, PWN, Warszawa, 1977. Zbl0684.54001MR0500780
  10. Skljarenko E.G., A theorem on maps, lowering dimension (in Russian), Bull. Acad. Polon. Sci., Ser. Mat. 10 (1962), 429-432. (1962) MR0149445

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