Decompositions of spaces determined by compact subsets

Author:
Yoshio Tanaka

Journal:
Proc. Amer. Math. Soc. **97** (1986), 549-555

MSC:
Primary 54D50; Secondary 54E60

MathSciNet review:
840644

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let be a -space, and let be a closed cover of (locally) compact subsets of . Then is decomposed into a closed discrete subset and a locally compact subset if is dominated by , or has the weak topology with respect to a point-countable cover . Here, a cover of a space is point-countable if every point is in at most countably many elements of the cover.

**[1]**A. V. Arhangel'skiĭ,*On a class of spaces containing all metric and all locally bicompact spaces*, Soviet Math. Dokl.**4**(1963), 1051-1055.**[2]**-,*Some types of factor mappings, and the relations between classes of topological spaces*, Soviet Math. Dokl.**4**(1963), 1726-1729.**[3]**A. V. Arhangel′skiĭ,*Bicompact sets and the topology of spaces.*, Trudy Moskov. Mat. Obšč.**13**(1965), 3–55 (Russian). MR**0195046****[4]**A. V. Arkhangel′skiĭ and V. I. Ponomarev,*Fundamentals of general topology*, Mathematics and its Applications, D. Reidel Publishing Co., Dordrecht, 1984. Problems and exercises; Translated from the Russian by V. K. Jain; With a foreword by P. Alexandroff [P. S. Aleksandrov]. MR**785749****[5]**V. V. Filippov,*On feathered paracompacta*, Soviet Math. Dokl.**9**(1968), 161-164.**[6]**-,*On factor**-mappings*, Soviet Math. Dokl.**9**(1968), 1055-1057.**[7]**Ben Fitzpatrick Jr.,*Some topologically complete spaces*, General Topology and Appl.**1**(1971), no. 1, 101–103; errata, MR 45, p. 1978. MR**0298660****[8]**G. Gruenhage, E. Michael, and Y. Tanaka,*Spaces determined by point-countable covers*, Pacific J. Math.**113**(1984), no. 2, 303–332. MR**749538****[9]**N. Lašnev,*Continuous decompositions and closed mappings of metric spaces*, Soviet Math. Dokl.**6**(1965), 1504-1506.**[10]**Ernest Michael,*Continuous selections. I*, Ann. of Math. (2)**63**(1956), 361–382. MR**0077107****[11]**E. Michael,*A note on closed maps and compact sets*, Israel J. Math.**2**(1964), 173–176. MR**0177396****[12]**E. A. Michael,*A quintuple quotient quest*, General Topology and Appl.**2**(1972), 91–138. MR**0309045****[13]**Kiiti Morita,*On spaces having the weak topology with respect to closed coverings*, Proc. Japan Acad.**29**(1953), 537–543. MR**0062423****[14]**Kiiti Morita,*On closed mappings*, Proc. Japan Acad.**32**(1956), 539–543. MR**0085494****[15]**Kiiti Morita,*On decomposition spaces of locally compact spaces*, Proc. Japan Acad.**32**(1956), 544–548. MR**0085495****[16]**V. Ponomarev,*Axioms of countability and continuous mappings*, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Phys.**8**(1960), 127–134 (Russian, with English summary). MR**0116314****[17]**Yoshio Tanaka,*Point-countable 𝑘-systems and products of 𝑘-spaces*, Pacific J. Math.**101**(1982), no. 1, 199–208. MR**671852****[18]**Yoshio Tanaka and Hao Xuan Zhou,*Spaces dominated by metric subsets*, Proceedings of the 1984 topology conference (Auburn, Ala., 1984), 1984, pp. 149–163. MR**781558****[19]**J. H. C. Whitehead,*Combinatorial homotopy. I*, Bull. Amer. Math. Soc.**55**(1949), 213–245. MR**0030759**, 10.1090/S0002-9904-1949-09175-9

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
54D50,
54E60

Retrieve articles in all journals with MSC: 54D50, 54E60

Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0840644-9

Keywords:
Weak topology,
-space,
-space,
locally compact space,
quotient map,
pseudo-open map,
closed map

Article copyright:
© Copyright 1986
American Mathematical Society