Preservation of topological properties under extensions of topologies

Abstract

Let (X, T) be a topological space and let A be a subset of X. The topology generated by T and A is defined to be the simple extension of the topology T to the set A. Sone elementary properties of simple extensions of topologies are examined. Necessary and sufficient conditions are given for the preservation of various hereditary and weakly hereditary topological properties under a simple extension of the topology. It is shown that connectedness is preserved if A is a dense set. The use of simple extensions in the construction of certain types of counterexamples is discussed. The more general concepts of finite and infinite extensions of topologies are defined to be the topology generated by T and a given collection {A_alpha | alpha in Gamma} of subsets of x. Necessary and sufficient conditions are given for the preservation of various topological properties under finite extensions of topologies. It is shown that every open set in an infinite extension topology can be expressed as a union of sets, each of which is open in some related finite extension topology. By placing various restrictions on the collection {A_alpha | alpha in Gamma}, it is shown that regularity, complete regularity, paracompactness, metrizability, normality and connectedness are preserved under infinite extensions of topologies.