Vol. 31, No. 2, 1969

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Minimal T0-spaces and minimal TD-spaces

Roland Edwin Larson

Vol. 31 (1969), No. 2, 451–458

The family of all topologies on a set is a complete, atomic lattice. There has been a considerable amount of interest in topologies which are minimal or maximal in this lattice with respect to certain topological properties. Given a topological property P, we say a topology is minimal P (maximal P) if every weaker (stronger) topology does not possess property P. A topological space (X,r ) is called a TD-space if and only if [x]floor, (the derived set of [x]) is a closed set for every x in X[1]. It is known that a space is TD if and only if for every x in X there exists an open set G and a closed set C such that [x] = G C[9]. The purpose of this paper is to characterize minimal T0 and minimal TD-spaces as follows: A T0-space is minimal T0 if and only if the family of open sets is nested and the complements of the point closures form a base for the topology. A TD-space is minimal TD if and only if the open sets are nested. These characterizations prove to be useful in gaining other results about minimal T0 and minimal TD-spaces.

Mathematical Subject Classification
Primary: 54.23
Received: 3 February 1969
Published: 1 November 1969
Roland Edwin Larson