A Note on Monotonic Ortho-Bases

Thomas M. Phillips
1977 Proceedings of the American Mathematical Society  
At the 1974 Topology Conference at Charlotte, North Carolina, Peter Nyikos introduced the concept of an ortho-base and announced that a 7"2 paracompact first-countable /8-space having an ortho-base is metrizable. The purpose of this paper is to introduce an obvious monotonie generalization of ortho-bases and to prove the following theorem. Theorem. If S is a regular T0 space having a monotonie ortho-base, then each of the following implies that S has a base of countable order: (1) 5 is
more » ... : (1) 5 is connected; (2) S is a ßc-space; (3) S is a first-countable monotonie ß-space. Nyikos' theorem is a corollary to (3) and Arhangel'ski?s theorem that a T2 paracompact space having a base of countable order is metrizable. An ortho-base for a space A1 is a base B for the topology of X such that if F C B, then either D F is open or D F = {x} and F is a local base at x. Every space having an ortho-base is orthocompact1 and for T0 developable spaces, the converse is true. However, the space of countable ordinals is an orthocompact space which does not have an ortho-base. The concept of an ortho-base was introduced by Nyikos in [5] where he announced [5, Theorem 3.1] that a T2 paracompact first countable /8-space2 having an ortho-base is metrizable. The purpose of this note is to show that a regular T0 first countable monotonie /8-space having a monotonie ortho-base (see definitions below) has a base of countable order thereby obtaining Nyikos' theorem as a corollary to the well-known theorem of Arhangel'skii that a T2 paracompact space having a base of countable order is metrizable. The primitive concepts of Wicke and Worrell are essential tools in the investigation of spaces having bases of countable order and it is assumed that the reader is acquainted with the terminology and techniques found in [7] and [8] . An example of the utility of these technical results is the following lemma, the proof of which is implicit in the works of these authors.
doi:10.2307/2042012 fatcat:p7njfgtzwverljvx4gxqgofgva