A note on monotonic ortho-bases

Author:
Thomas M. Phillips

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

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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 paracompact first-countable -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* *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* -space;

(3) *S is a first-countable monotonic* -space.

Nyikos' theorem is a corollary to (3) and Arhangel'skii's theorem that a paracompact space having a base of countable order is metrizable.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1977-0442898-4

Keywords:
Base of countable order,
-space,
monotonic -space,
monotonic ortho-base,
monotonically developable,
ortho-base

Article copyright:
© Copyright 1977
American Mathematical Society