The cohesive property
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- by Gerald Jungck PDF
- Proc. Amer. Math. Soc. 84 (1982), 138-142 Request permission
Abstract:
We introduce the concept of cohesive families of neighborhood bases. We thereby obtain conditions necessary and sufficient to ensure that a separable space be second countable, and sufficiency conditions for complete collectionwise normality. As by-products we obtain metrizability criteria. We prove, e.g., that a ${T_1}$ space is metrizable iff it has a refined development $\{ {G_n}:n \in N\}$ such that $\{ {B_p}:p \in X\}$ with ${B_p} = \{ {\text {St}}(p,{G_n}):n \in N\}$ is cohesive.References
- Francesca Cagliari and Marcello Cicchese, Continuity conditions and convergence properties in generalized metric spaces, Riv. Mat. Univ. Parma (4) 2 (1976), 329–336 (English, with Italian summary). MR 445463 H. Cook, Cartesian products and continuous semimetrics, Proceedings of Point Set Topology Conference (Editor, E. Grace), Arizona State Univ., Tempe, 1967. R. W. Heath, On certain first countable spaces, Topology Seminar, Wisconsin Univ. Press and Princeton Univ. Press, Princeton, N. J., 1966, pp. 103-113.
- Louis F. McAuley, A note on complete collectionwise normality and paracompactness, Proc. Amer. Math. Soc. 9 (1958), 796–799. MR 99647, DOI 10.1090/S0002-9939-1958-0099647-X
- Louis F. McAuley, A relation between perfect separability, completeness, and normality in semi-metric spaces, Pacific J. Math. 6 (1956), 315–326. MR 80907
- Lynn Arthur Steen and J. Arthur Seebach Jr., Counterexamples in topology, 2nd ed., Springer-Verlag, New York-Heidelberg, 1978. MR 507446
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 84 (1982), 138-142
- MSC: Primary 54E65
- DOI: https://doi.org/10.1090/S0002-9939-1982-0633295-9
- MathSciNet review: 633295