Algebras and Spaces of Dense Constancies

A DC-space (or space of dense constancies) is a Tychonoff space X such that for each f C(X) there is a family of open sets {U _i : i I}, the union of which is dense in X, such that f, restricted to each U _i , is constant. A number of characterizations of DC-spaces are given, which lead to an algebraic generalization of the concept, which, in turn, permits analysis of DC-spaces in the language of archimedean f-algebras. One is led naturally to the notion of an almost DC-space (in which the densely constant functions are dense), and it is shown that all metrizable spaces have this property.