| Title:
|
Existence of quasicontinuous selections for the space $2\sp {f R}$ (English) |
| Author:
|
Kupka, Ivan |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
121 |
| Issue:
|
2 |
| Year:
|
1996 |
| Pages:
|
157-163 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
The paper presents new quasicontinuous selection theorem for continuous multifunctions $F X \longrightarrow\Bbb R$ with closed values, $X$ being an arbitrary topological space. It is known that for $2^{\Bbb R}$ with the Vietoris topology there is no continuous selection. The result presented here enables us to show that there exists a quasicontinuous and upper$\langle$lower$\rangle$-semicontinuous selection for this space. Moreover, one can construct a selection whose set of points of discontinuity is nowhere dense. (English) |
| Keyword:
|
continuous multifunction |
| Keyword:
|
selection |
| Keyword:
|
quasicontinuity |
| MSC:
|
54C08 |
| MSC:
|
54C65 |
| idZBL:
|
Zbl 0863.54014 |
| idMR:
|
MR1400608 |
| DOI:
|
10.21136/MB.1996.126098 |
| . |
| Date available:
|
2009-09-24T21:17:47Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/126098 |
| . |
| Reference:
|
[1] R. Engelking R. V. Heath E. Michael: Topological well-ordering and continuous selections.Invent. Math. 6 (1968), 150-158. MR 0244959, 10.1007/BF01425452 |
| Reference:
|
[2] I. Kupka: Quasicontinuous selections for compact-valued multifunctions.Math. Slovaca 43 (1993), 69-75. Zbl 0784.54023, MR 1216269 |
| Reference:
|
[3] I. Kupka: Continuous multifunction from [-1,0] to R having noncontinuous selection.Publ. Math. (Submitted). |
| Reference:
|
[4] K. Kuratowski: Topologie I.PWN Warszawa, 1952. |
| Reference:
|
[5] N. Levine: Semi-open sets and semi-continuity in topological spaces.Amer. Math. Monthly 70 (1963), 36-41. Zbl 0113.16304, MR 0166752, 10.1080/00029890.1963.11990039 |
| Reference:
|
[6] M. Matejdes: Sur les sélecteurs des multifonctions.Math. Slovaca 37 (1987), 111-124. Zbl 0629.54013, MR 0899022 |
| Reference:
|
[7] M. Matejdes: On selections of multifunctions.Math. Bohem. 118 (1993), 255-260. Zbl 0785.54022, MR 1239120 |
| Reference:
|
[8] M. Matejdes: Quasi-continuous and cliquish selections of multifunctions on product spaces.Real Anal. Exchange. To appear. Zbl 0782.54019, MR 1205513 |
| Reference:
|
[9] E. Michael: Continuous selections I.Ann. of Math. 63 (1956), 361-382. Zbl 0071.15902, MR 0077107, 10.2307/1969615 |
| Reference:
|
[10] E. Michael: Continuous selections II.Ann. of Math. 64 (1956), 562-580. Zbl 0073.17702, MR 0080909, 10.2307/1969603 |
| Reference:
|
[11] S. B. Nadler: Hyperspaces of sets.Marcel Dekker, Inc., New York and Bassel, 1978. Zbl 0432.54007, MR 0500811 |
| Reference:
|
[12] T. Neubrunn: Quasi-continuity.Real Anal. Exchange H (1988-89), 259-306. MR 0995972 |
| Reference:
|
[13] A. Neubrunnová: On certain generalizations of the notion of continuity.Mat. Časopis 23 (1973), 374-380. MR 0339051 |
| . |
