Spaces in which compact sets have countable local bases

Author:
Ralph R. Sabella

Journal:
Proc. Amer. Math. Soc. **48** (1975), 499-504

MSC:
Primary 54E35

MathSciNet review:
0370522

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Abstract: -spaces and coconvergent spaces are examples of spaces in which compact sets have countable local bases (-spaces). Among the results related to -spaces given in this paper is a sufficient condition under which such spaces are coconvergent. In relation to a question posed by F. B. Jones, it is shown that a topological property sufficient for semimetrizable spaces to be developable is that they be coconvergent. Coconvergence implies metrizability in stratifiable spaces; it is shown in this paper that a -space is metrizable if there exists a stratification satisfying a nesting-like condition.

**[1]**C. E. Aull,*Closed set countability axioms*, Nederl. Akad. Wetensch. Proc. Ser. A 69=Indag. Math.**28**(1966), 311–316. MR**0199833****[2]**Carlos J. R. Borges,*On stratifiable spaces*, Pacific J. Math.**17**(1966), 1–16. MR**0188982****[3]**Jack G. Ceder,*Some generalizations of metric spaces*, Pacific J. Math.**11**(1961), 105–125. MR**0131860****[4]**Geoffrey D. Creede,*Concerning semi-stratifiable spaces*, Pacific J. Math.**32**(1970), 47–54. MR**0254799****[5]**R. W. Heath,*On certain first-countable spaces*, Ann. of Math. Studies, no. 60, Princeton Univ. Press, Princeton, N.J., 1965, pp. 103-113.**[6]**Ralph R. Sabella,*Convergence properties of neighboring sequences*, Proc. Amer. Math. Soc.**38**(1973), 405–409. MR**0312479**, 10.1090/S0002-9939-1973-0312479-8**[7]**-,*Properties of neighboring sequences in stratifiable spaces*, Proc. Amer. Math. Soc. (submitted).

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0370522-6

Keywords:
Open neighborhood assignments,
-linked sequences,
coconvergent,
contraconvergent,
stratifiable spaces,
Nagata spaces

Article copyright:
© Copyright 1975
American Mathematical Society