Dimension-theoretic properties of completions

Abstract:

In this paper we extend some previous work from situations involving countable collections of subsets to those concerning locally finite collections. An example of the results obtained here is a theorem which asserts that corresponding to any locally finite collection of finite-dimensional closed subsets of a metric space $X$ there exists a completion of $X$ in which taking the closure of any member of the given collection does not raise dimension. The basic technique employed in each of the proofs is similar; a topologically equivalent metric is introduced (one which is strongly dependent upon the given locally finite collection), and the desired completion is then taken with respect to this new metric.