A note on monotonic ortho-bases
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- by Thomas M. Phillips
- Proc. Amer. Math. Soc. 65 (1977), 150-154
- DOI: https://doi.org/10.1090/S0002-9939-1977-0442898-4
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Abstract:
At the 1974 Topology Conference at Charlotte, North Carolina, Peter Nyikos introduced the concept of an ortho-base and announced that a ${T_2}$ paracompact first-countable $\beta$-space having an ortho-base is metrizable. The purpose of this paper is to introduce an obvious monotonic generalization of ortho-bases and to prove the following theorem. Theorem. If S is a regular ${T_0}$ space having a monotonic ortho-base, then each of the following implies that S has a base of countable order: (1) S is connected; (2) S is a ${\beta _c}$-space; (3) S is a first-countable monotonic $\beta$-space. Nyikos’ theorem is a corollary to (3) and Arhangel’skiǐ’s theorem that a ${T_2}$ paracompact space having a base of countable order is metrizable.References
- Dennis K. Burke, A nondevelopable locally compact Hausdorff space with a $G\delta$-diagonal, General Topology and Appl. 2 (1972), 287–291. MR 319156
- J. Chaber, On point-countable collections and monotonic properties, Fund. Math. 94 (1977), no. 3, 209–219. MR 451206, DOI 10.4064/fm-94-3-209-219
- Peter Fletcher and William F. Lindgren, Orthocompactness and strong Čech completeness in Moore spaces, Duke Math. J. 39 (1972), 753–766. MR 312472
- R. E. Hodel, Moore spaces and $w$ $\Delta$-spaces, Pacific J. Math. 38 (1971), 641–652. MR 307169
- Peter J. Nyikos, Some surprising base properties in 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. 427–450. MR 0367940
- Howard H. Wicke, Open continuous images of certain kinds of $M$-spaces and completeness of mappings and spaces, General Topology and Appl. 1 (1971), no. 1, 85–100. MR 282348
- H. H. Wicke and J. M. Worrell Jr., Completeness and topologically uniformizing structures, TOPO 72—general topology and its applications (Proc. Second Pittsburgh Internat. Conf., Pittsburgh, Pa., 1972; dedicated to the memory of Johannes H. de Groot), Lecture Notes in Math., Vol. 378, Springer, Berlin, 1974, pp. 557–585. MR 0375241
- 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., Topological completeness of first countable Hausdorff spaces. I, Fund. Math. 75 (1972), no. 3, 209–222. MR 309078, DOI 10.4064/fm-75-3-209-222
Bibliographic Information
- © Copyright 1977 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 65 (1977), 150-154
- MSC: Primary 54E99
- DOI: https://doi.org/10.1090/S0002-9939-1977-0442898-4
- MathSciNet review: 0442898