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A functional characterization of primitive base

Author: Howard H. Wicke
Journal: Proc. Amer. Math. Soc. 72 (1978), 355-361
MSC: Primary 54E99; Secondary 54A99
MathSciNet review: 507338
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Abstract: All published formulations of the concept of primitive base involve the concept of well ordering in a prominent way. This paper presents various conditions, not involving well ordering, on a function $ h:N \times X \to \tau $ such that if a space $ (X,\tau )$ has a function satisfying one of these conditions, then it may be proved (using the axiom of choice) that it has a primitive base. These conditions are used in some characterizations of base of countable order. Examples show nonequivalence of the conditions with primitive base if the axiom of choice does not hold.

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  • [1] D. K. Burke and D. J. Lutzer, Recent advances in the theory of generalized metric spaces, Topology, Proc. of the Memphis State University Conference, Marcel Dekker, New York, 1976, pp. 1-70. MR 55 #1318. MR 0428293 (55:1318)
  • [2] P. Fletcher and W. F. Lindgren, $ \theta $-spaces, General Topology and Appl. (to appear). MR 0500849 (58:18362)
  • [3] R. Heath, Arcwise connectedness in semi-metric spaces, Pacific J. Math. 12 (1965), 91-96. MR 0166759 (29:4032)
  • [4] -, On open mappings and certain spaces satisfying the first countability axiom, Fund. Math. 62 (1965), 91-96. MR 0179763 (31:4006)
  • [5] R. Hodel, Spaces defined by sequences of open covers which guarantee that certain sequences have cluster points, Duke Math. J. 39 (1972), 253-263. MR 0293580 (45:2657)
  • [6] H. H. Wicke, Open continuous images of certain kinds of M-spaces and completeness of mappings and spaces, General Topology and Appl. 1 (1971), 85-100. MR 0282348 (43:8060)
  • [7] 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 51 #11438. MR 0375242 (51:11438)
  • [8] -, A characterization of primitive bases, Proc. Amer. Math. Soc. 50 (1975), 443-450. MR 0425922 (54:13872)
  • [9] -, 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)
  • [10] -, The hereditary Lindelöf property, primitive structures, and separable metrizability, Notices Amer. Math. Soc. 22 (1975), A-425, Abstract 723-G10.
  • [11] -, On spaces which are scattered with respect to collections of sets, Topology Proceedings 2 (1977), 281-307. MR 540612 (80k:54052)

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Keywords: Primitive base, base of countable order, $ \theta $-space, first countable function, well ordering
Article copyright: © Copyright 1978 American Mathematical Society

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