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.
