Complete exhaustive sieves and games
HTML articles powered by AMS MathViewer
- by Rastislav Telgársky and Howard H. Wicke PDF
- Proc. Amer. Math. Soc. 102 (1988), 737-744 Request permission
Abstract:
Properties of completely regular spaces with complete exhaustive sieves are studied using the equivalent notion of partition complete spaces and associated games. Among others the following results are proved. (1) If $X$ is the intersection of countably many partition complete subsets of $Y$, then $X$ is partition complete. (2) If $X$ is $K$-scattered, where $K$ is the class of all partition complete spaces, then $X$ is partition complete. (3) If $X$ is a primitive set in a $C$-scattered space $Y$, then $X$ is the intersection of countably many $C$-scattered subsets of $Y$. (4) The partition completeness is perfect and preserved by open maps.References
- H. J. K. Junnila, J. C. Smith, and R. Telgársky, Closure-preserving covers by small sets, Topology Appl. 23 (1986), no. 3, 237–262. MR 858333, DOI 10.1016/0166-8641(85)90042-2
- E. Michael, A note on completely metrizable spaces, Proc. Amer. Math. Soc. 96 (1986), no. 3, 513–522. MR 822451, DOI 10.1090/S0002-9939-1986-0822451-6
- Thomas M. Phillips, Extending continuous functions, Topology (Proc. Ninth Annual Spring Topology Conf., Memphis State Univ., Memphis, Tenn., 1975) Lecture Notes in Pure and Appl. Math., Vol. 24, Dekker, New York, 1976, pp. 229–237. MR 0454906
- S. Purisch, The orderability and closed images of scattered spaces, Trans. Amer. Math. Soc. 320 (1990), no. 2, 713–725. MR 989584, DOI 10.1090/S0002-9947-1990-0989584-0
- Rastislav Telgársky, $C$-scattered and paracompact spaces, Fund. Math. 73 (1971/72), no. 1, 59–74. MR 295293, DOI 10.4064/fm-73-1-59-74
- Rastislav Telgársky, Spaces defined by topological games, Fund. Math. 88 (1975), no. 3, 193–223. MR 380708, DOI 10.4064/fm-88-3-193-223
- Rastislav Telgársky, On sieve-complete and compactlike spaces, Topology Appl. 16 (1983), no. 1, 61–68. MR 702620, DOI 10.1016/0166-8641(83)90007-X H. H. Wicke, Complete exhaustive sieves, Topology Proc. (to appear).
- H. H. Wicke and J. M. Worrell Jr., Primitive structures in general topology, Studies in topology (Proc. Conf., Univ. North Carolina, Charlotte, N.C., 1974; dedicated to Math. Sect. Polish Acad. Sci.), Academic Press, New York, 1975, pp. 581–599. MR 0375242
- H. H. Wicke and J. M. Worrell Jr., A characterization of spaces having bases of countable order in terms of primitive bases, Canadian J. Math. 27 (1975), no. 5, 1100–1109. MR 385817, DOI 10.4153/CJM-1975-115-1
Additional Information
- © Copyright 1988 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 102 (1988), 737-744
- MSC: Primary 54E99; Secondary 54D99
- DOI: https://doi.org/10.1090/S0002-9939-1988-0929013-2
- MathSciNet review: 929013