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A note on monotonic ortho-bases

Author: Thomas M. Phillips
Journal: Proc. Amer. Math. Soc. 65 (1977), 150-154
MSC: Primary 54E99
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 $ {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'skii's theorem that a $ {T_2}$ paracompact space having a base of countable order is metrizable.

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Keywords: Base of countable order, $ \beta $-space, monotonic $ \beta $-space, monotonic ortho-base, monotonically developable, ortho-base
Article copyright: © Copyright 1977 American Mathematical Society

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