Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Journals Home Search My Subscriptions Subscribe
Previous issue | This issue | Most recent issue | All issues (1950–Present) | Next issue | Previous article | Articles in press | Recently published articles | Next article
Request Permissions
 
 

 

Complete exhaustive sieves and games


Authors: Rastislav Telgársky and Howard H. Wicke
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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [JST] H. J. K. Junnila, J. C. Smith and R. Telgársky, Closure-preserving covers by small sets, Topology and Appl. 23 (1986), 237-262. MR 858333 (87m:54063)
  • [M] E. Michael, A note on completely metrizable spaces, Proc. Amer. Math. Soc. 96 (1986), 512-522. MR 822451 (87f:54045)
  • [Ph] T. M. Phillips, Extending continuous functions, Topology Proc. (Memphis State Univ. Conf.) Dekker, New York, 1976, pp. 229-237. MR 0454906 (56:13149)
  • [Pu] S. Purish, The orderability and closed images of scattered spaces, preprint, 1986. MR 989584 (90k:54044)
  • [T$ _{1}$] R. Telgársky, $ C$-scattered and paracompact spaces, Fund. Math. 73 (1971), 59-74. MR 0295293 (45:4361)
  • [T$ _{2}$] -, Space defined by topological games, Fund. Math. 88 (1975), 193-223. MR 0380708 (52:1605)
  • [T$ _{3}$] -, On sieve-complete and compact-like spaces, Topology and Appl. 16 (1983), 61-68. MR 702620 (84h:54024)
  • [W] H. H. Wicke, Complete exhaustive sieves, Topology Proc. (to appear).
  • [WW$ _{1}$] H. H. Wicke and J. M. Worrell, Jr., Primitive structures in general topology, Studies in Topology, Academic Press, New York, 1975, pp. 581-599. MR 0375242 (51:11438)
  • [WW$ _{2}$] -, A characterization of spaces having bases of countable order in terms of primitive bases, Canad. J. Math. 27 (1975), 1100-1109. MR 0385817 (52:6676)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 54E99, 54D99

Retrieve articles in all journals with MSC: 54E99, 54D99

Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0929013-2
Keywords: Partition complete space, sieve complete space, $ K$-scattered space, exhaustive sieve, open presieve, primitive set, perfect mapping, open mapping, topological game
Article copyright: © Copyright 1988 American Mathematical Society

American Mathematical Society