A countably compact -space need not be countably

Author:
Roy C. Olson

Journal:
Proc. Amer. Math. Soc. **62** (1977), 144-148

MSC:
Primary 54D50; Secondary 54G20

DOI:
https://doi.org/10.1090/S0002-9939-1977-0493974-1

MathSciNet review:
0493974

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Abstract: An example is given of a countably compact *k*'-space that is not countably bi-*k*. Interest for this example arises from a recent paper of Michael, Olson, and Siwiec and from a 1972 paper of E. Michael, both of which discuss mapping characterizations of a range space. The construction of the example assumes the continuum hypothesis.

**[1]**A. Arhangel′skiĭ,*Some types of factor mappings and the relations between classes of topological spaces*, Dokl. Akad. Nauk SSSR**153**(1963), 743–746 (Russian). MR**0158362****[2]**A. V. Arhangel′skiĭ,*Bicompact sets and the topology of spaces.*, Trudy Moskov. Mat. Obšč.**13**(1965), 3–55 (Russian). MR**0195046****[3]**A. V. Arhangel′skiĭ,*Frequency spectrum of a topological space and classification of spaces*, Dokl. Akad. Nauk SSSR**206**(1972), 265–268 (Russian). MR**0394575****[4]**Leonard Gillman and Meyer Jerison,*Rings of continuous functions*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0116199****[5]**E. Michael,*ℵ₀-spaces*, J. Math. Mech.**15**(1966), 983–1002. MR**0206907****[6]**E. A. Michael,*A quintuple quotient quest*, General Topology and Appl.**2**(1972), 91–138. MR**0309045****[7]**E. Michael, R. C. Olson, and F. Siwiec,*𝐴-spaces and countably bi-quotient maps*, Dissertationes Math. (Rozprawy Mat.)**133**(1976), 43. MR**0418023****[8]**Roy C. Olson,*Bi-quotient maps, countably bi-sequential spaces, and related topics*, General Topology and Appl.**4**(1974), 1–28. MR**0365463****[9]**R. C. Olson,*Some results from 𝐴-spaces*, Set-theoretic topology (Papers, Inst. Medicine and Math., Ohio Univ., Athens, Ohio, 1975–1976) Academic Press, New York, 1977, pp. 307–312. MR**0442881****[10]**Frank Siwiec,*Sequence-covering and countably bi-quotient mappings*, General Topology and Appl.**1**(1971), no. 2, 143–154. MR**0288737**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0493974-1

Keywords:
*k*'-space,
strongly *k*'-space,
countably bi-*k*-space,
singly bi-*k*-space

Article copyright:
© Copyright 1977
American Mathematical Society