A note on monotonic ortho-bases

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 {T_2} paracompact first-countable β \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 {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 β c {\beta _c} -space; (3) S is a first-countable monotonic β \beta -space. Nyikos’ theorem is a corollary to (3) and Arhangel’skiǐ’s theorem that a T 2 {T_2} paracompact space having a base of countable order is metrizable.