| Title:
|
A note on splittable spaces (English) |
| Author:
|
Tkachuk, Vladimir V. |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
33 |
| Issue:
|
3 |
| Year:
|
1992 |
| Pages:
|
551-555 |
| . |
| Category:
|
math |
| . |
| 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. (English) |
| Keyword:
|
splittable |
| Keyword:
|
polyhedron |
| Keyword:
|
dimension |
| MSC:
|
54A25 |
| MSC:
|
54D99 |
| idZBL:
|
Zbl 0769.54004 |
| idMR:
|
MR1209296 |
| . |
| Date available:
|
2009-01-08T17:58:07Z |
| Last updated:
|
2012-04-30 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/118522 |
| . |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[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 |
| Reference:
|
[6] Mazurkiewicz S.: Sur les problèmes $\kappa $ et $\lambda $ de Urysohn.Fund. Math. 10 (1927), 311-319. |
| Reference:
|
[7] Tkachuk V.: Remainders over discrete spaces - some applications (in Russian).Vestnik MGU, Mat., Mech., no. 4, 1990, 18-21. MR 1086601 |
| Reference:
|
[8] Malyhin V.I.: $\beta N$ is prime.Bull. Acad. Polon. Sci., Ser. Mat. 27 (1979), 295-297. Zbl 0433.54015, MR 0552052 |
| Reference:
|
[9] Engelking R.: General Topology.PWN, Warszawa, 1977. Zbl 0684.54001, MR 0500780 |
| Reference:
|
[10] Skljarenko E.G.: A theorem on maps, lowering dimension (in Russian).Bull. Acad. Polon. Sci., Ser. Mat. 10 (1962), 429-432. MR 0149445 |
| . |
