Characterizations of paracompact-like properties by means of set-valued semi-continuous selections

Author:
Kazumi Miyazaki

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2777-2782

MSC (2000):
Primary 54B20, 54C65

Published electronically:
April 24, 2001

MathSciNet review:
1838802

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

We give a characterization of countably paracompact and collectionwise normal spaces by means of set-valued semi-continuous selections. This provides a positive answer to a problem of V. Gutev.

**1.**J. Chaber, M. M. Čoban, and K. Nagami,*On monotonic generalizations of Moore spaces, Čech complete spaces and 𝑝-spaces*, Fund. Math.**84**(1974), no. 2, 107–119. MR**0343244****2.**M. M. Čoban,*Multi-valued mappings and Borel sets. I, II*, Trudy Moskov. Mat. Obšč.**22**(1970), 229–250; ibid. 23 (1970), 277–301. MR**0372812****3.**S. Ĭ. Nedev and M. M. Čoban,*Factorization theorems for multivalued mappings, multivalued cross sections and topological dimension*, Math. Balkanica**4**(1974), 457–460 (Russian). Papers presented at the Fifth Balkan Mathematical Congress (Belgrade, 1974). MR**0388328****4.**C. H. Dowker,*Homotopy extension theorems*, Proc. London Math. Soc. (3)**6**(1956), 100–116. MR**0073982****5.**Ryszard Engelking,*General topology*, 2nd ed., Sigma Series in Pure Mathematics, vol. 6, Heldermann Verlag, Berlin, 1989. Translated from the Polish by the author. MR**1039321****6.**M. K. Fort Jr.,*A unified theory of semi-continuity*, Duke Math. J.**16**(1949), 237–246. MR**0030118****7.**V. Gutev,*Generic extensions of finite-valued u.s.c. selections*, Topology and Appl.**104**(2000), 101-118. CMP**2001:01****8.**Valentin G. Gutev,*Weak factorizations of continuous set-valued mappings*, Topology Appl.**102**(2000), no. 1, 33–51. MR**1739262**, 10.1016/S0166-8641(98)00139-4**9.**-,*Selections of set-valued mappings and hyperspace topologies*, (unpublished).**10.**Valentin Gutev and Tsugunori Nogura,*Selections for Vietoris-like hyperspace topologies*, Proc. London Math. Soc. (3)**80**(2000), no. 1, 235–256. MR**1719152**, 10.1112/S0024611500012107**11.**Ernest Michael,*Topologies on spaces of subsets*, Trans. Amer. Math. Soc.**71**(1951), 152–182. MR**0042109**, 10.1090/S0002-9947-1951-0042109-4**12.**E. Michael,*A theorem on semi-continuous set-valued functions*, Duke Math. J**26**(1959), 647–651. MR**0109343****13.**Ernest Michael,*Continuous selections. I*, Ann. of Math. (2)**63**(1956), 361–382. MR**0077107****14.**E. Michael,*Complete spaces and tri-quotient maps*, Illinois J. Math.**21**(1977), no. 3, 716–733. MR**0467688****15.**Stoyan Ĭ. Nedev,*Selection and factorization theorems for set-valued mappings*, Serdica**6**(1980), no. 4, 291–317 (1981). MR**644284**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2000):
54B20,
54C65

Retrieve articles in all journals with MSC (2000): 54B20, 54C65

Additional Information

**Kazumi Miyazaki**

Affiliation:
Department of Mathematical Sciences, Faculty of Sciences, Ehime University, Matsuyama 790-8577, Japan

Email:
BZQ22206@nifty.ne.jp

DOI:
https://doi.org/10.1090/S0002-9939-01-06204-9

Keywords:
Hyperspace,
selection,
semi-continuous

Received by editor(s):
May 12, 1999

Received by editor(s) in revised form:
December 12, 1999

Published electronically:
April 24, 2001

Communicated by:
Alan Dow

Article copyright:
© Copyright 2001
American Mathematical Society