Spaces in which compact sets have countable local bases

Abstract

D1-spaces and coconvergent spaces are examples of spaces in which compact sets have countable local bases (D0-spaces). Among the results related to D0-spaces given in this paper is a sufficient condition under which such spaces are coconvergent. In relation to a question posed by F. B. Jones, it is shown that a topological property sufficient for semimetrizable spaces to be developable is that they be coconvergent. Coconvergence implies metrizability in stratifiable spaces; it is shown in this paper that a D0-space is metrizable if there exists a stratification satisfying a nesting-like condition. An open neighborhood assignment (ONA) is a function U: x N U I(x): x 6 such that x E U(x, n) _ Un(x), where is a topological space, N is the set of natural numbers and /q(x) is the open neighborhood system of x. If U is an ONA then the sequence yn} is U-linked to lxn) if Yn E n for all n. Using the notation CplxnI for the set of cluster points of ixn }, we define a space to be coconvergent (contraconvergent) if CplxnI C Cp'yn (Cp'xn} I Cp'yn}) whenever lyn } is U-linkedto lxn . If on there is an ONA U having some property P, we shall say X is P or IU is P. If x0 is a limit point of xn}, then A(xny x0) _ Ixk: k 0, 1, 2, ...I. Coconvergence can be characterized in terms of countably based compact sets. From [6] we have Proposition 1. is coconvergent iff there is an ONA U on such that for any compact set K and open R containing K there is a k E N for which UUJk(x): x E K} C R. Received by the editors October 10, 1973 and, in revised form, January 31, 1974. AMS (MOS) subject classifications (1970). Primary 54E35; Secondary 54D99.