| Title:
|
On $\pi$-metrizable spaces, their continuous images and products (English) |
| Author:
|
Stover, Derrick |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
50 |
| Issue:
|
1 |
| Year:
|
2009 |
| Pages:
|
153-162 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
A space $X$ is said to be $\pi$-metrizable if it has a $\sigma$-discrete $\pi$-base. The behavior of $\pi$-metrizable spaces under certain types of mappings is studied. In particular we characterize strongly $d$-separable spaces as those which are the image of a $\pi$-metrizable space under a perfect mapping. Each Tychonoff space can be represented as the image of a $\pi$-metrizable space under an open continuous mapping. A question posed by Arhangel'skii regarding if a $\pi$-metrizable topological group must be metrizable receives a negative answer. (English) |
| Keyword:
|
$\pi$-metrizable |
| Keyword:
|
weakly $\pi$-metrizable |
| Keyword:
|
$\pi$-base |
| Keyword:
|
$\sigma$-discrete $\pi$-base |
| Keyword:
|
$\sigma$-disjoint $\pi$-base |
| Keyword:
|
$d$-separable |
| MSC:
|
54B10 |
| MSC:
|
54C10 |
| MSC:
|
54D70 |
| idZBL:
|
Zbl 1212.54033 |
| idMR:
|
MR2562812 |
| . |
| Date available:
|
2009-08-18T12:24:01Z |
| Last updated:
|
2013-09-22 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/133423 |
| . |
| Reference:
|
[1] Arhangel'skii A.V.: Topological invariants in algebraic environment.Recent Progress in General Topology, II, North-Holland, Amsterdam, 2002, pp. 1--57. Zbl 1030.54026, MR 1969992 |
| Reference:
|
[2] Arhangel'skii A.V.: $d$-separable spaces.Seminar on General Topology, Moscow, 1981, pp. 3--8. MR 0656944 |
| Reference:
|
[3] Davis S.: Topology.McGraw-Hill, New York, 2004. Zbl 1142.20020 |
| Reference:
|
[4] Engelking R.: General Topology.Heldermann, Berlin, 1989. Zbl 0684.54001, MR 1039321 |
| Reference:
|
[5] Fearnley D.: A Moore space with a $\sigma$-discrete $\pi$-base which cannot be densely embedded in any Moore space with the Baire property.Proc. Amer. Math. Soc. 127 (1999), 3095--3100. Zbl 0992.54026, MR 1605960, 10.1090/S0002-9939-99-04876-5 |
| Reference:
|
[6] Isbell J.: Uniform Spaces.American Mathematical Society, Providence, Rhode Island, 1964. Zbl 0124.15601, MR 0170323 |
| Reference:
|
[7] Ponomarev V.: On the absolute of a topological space.Dokl. Akad. Nauk SSSR 149 26--29 (1963). MR 0157355 |
| Reference:
|
[8] White H.E.: First countable spaces that have countable pseudo-bases.Canad. Math. Bull. 21 103--112 (1978). MR 0482615, 10.4153/CMB-1978-016-5 |
| . |
