A note on monotonic ortho-bases

Phillips, Thomas M.

doi:10.1090/S0002-9939-1977-0442898-4

A note on monotonic ortho-bases
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Proc. Amer. Math. Soc. 65 (1977), 150-154 Request permission

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.

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