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Product Maps and Countable Paracompactness

Published online by Cambridge University Press:  20 November 2018

Harold W. Martin
Harold W. Martin*
Affiliation:
University of Pittsburgh, Pittsburgh, Pennsylvania
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In all that follows, we let S denote the space {0, 1, 1/2, … , 1/n, …} with the relative usual topology and i : S → S denote the identity map on S. In this note, by a map or mapping we always mean a continuous surjection. A map f : X → Y is said to be hereditarily quotient if y ∊ int f(V) whenever V is open in X and f-1(y) ⊂ V. E. Michael has defined a map f : X → Y to be bi-quotient if whenever is a collection of open sets in X which covers f-1(y), there exists finitely many f(V), with V ∊ , which cover some neighbourhood of y.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 24 , Issue 6 , December 1972 , pp. 1187 - 1190
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Dowker, C. H., On countably paracompact spaces, Can. J. Math. 3 (1951), 219–224.Google Scholar
2. Dugundji, J., Topology (Allyn and Bacon, Boston, 1966).Google Scholar
3. Franklin, S. P., Spaces in which sequences suffice, Fund. Math. 57 (1965), 107–116.Google Scholar
4. Franklin, S. P., Spaces in which sequence suffice. II, Fund. Math. 61 (1967), 51–56.Google Scholar
5. Harley, P., On countably paracompact spaces and closed maps (to appear).Google Scholar
6. Henriksen, M. and Isbell, J. R., Some properties of compactifications, Duke Math. J. 25 (1958), 83–105.Google Scholar
7. Ishikawa, T., On countably paracompact space, Proc. Japan Acad. 31 (1955), 686–687.Google Scholar
8. Krajewski, L., On expanding locally finite collections, Can. J. Math. 23 (1971), 58–68.Google Scholar
9. Michael, E., A note on closed maps and compact sets, Israel J. Math. 2 (1964), 173–176.Google Scholar
10. Michael, E., Bi-quotient maps and cartesian products of quotient maps, Ann. Inst. Fourier (Grenoble) 18 (1968), 287–302.Google Scholar
11. Noble, N., Products with closed projections, Trans. Amer. Math. Soc. 160 (1971), 169–183.Google Scholar
12. Whyburn, G. T., Accessibility spaces, Proc. Amer. Math. Soc. 24 (1970), 181–185.Google Scholar
13. Zenor, P., On countable paracompactness and normality, Prace Mat. 13 (1969), 23–32.Google Scholar