On the representation of metric spaces

G.J. Logan
1979 Bulletin of the Australian Mathematical Society  
A closure algebra is a set X with a closure operator C defined on it. It is possible to construct a topology x on M v , the family of maximal, proper, closed subsets of I , and then to examine the relationship between the algebraic structure of {X, C) and the topological structure of the dual space [My, x) . This paper describes the algebraic conditions which are necessary and sufficient for the dual space to be separable metric and metric respectively. In Logan [3] a method was given for
more » ... enting any T -space as a particular kind of dual space of a closure algebra. The present paper improves the manner in which the closure algebra is constructed, and specializes the result to separable metric spaces and to metric spaces in general. The results seem to be of some interest because they offer a characterization of metric and separable metric spaces which to my knowledge is completely new. The notation, results, and references in Logan [3], [4] are presupposed: in particular if (X, C) is a closure algebra, then M is the family of maximal consistent sets; 5 : P{X) •* P[M Y ) , defined by si S{A) = {A € M x : A c A} will satisfy A c B =» S(B) c S(A) , and
doi:10.1017/s0004972700011072 fatcat:kmmetlglkvfi3kbw3l52wmn36q