Extension of continuous functions into uniform spaces

Abstract

ABSTRACT. Let X be a dense subspace of a topological space X, let Y be a uniformizable space, and let /: X -► Y a continuous map. In this paper we study the problem of the existence of a continuous extension of / to the space T. Thus we generalize basic results of Taimanov, Engelking and BlefkoMrówka on extension of continuous functions. As a consequence, if D is a nest generated intersection ring on X, we obtain a necessary and sufficient condition for the existence of a continuous extension to v(X, D), of a continuous function over X, when the range of the map is a uniformizable space, and we apply this to realcompact spaces. Finally, we suppose each point of T\X has a countable neighbourhood base, and we obtain a generalization of a theorem by Blair, herewith giving a solution to a question proposed by Blair.In the sequel the word space will designate a topological space. By pX we shall denote a uniform space, p being a collection of covers in the sense of Tukey [13] and Isbell [10]. 7/iX will mean the completion of the space pX. Let p be a uniformity on X and let m be an infinite'cardinal number, then pm will denote the uniformity on X generated by the //-covers of power < m (we only consider p and m for which pm is definable; see [6, 10 and 14]). The compact uniform space ipx0X is called the Samuel compactification of pX, denoted by spX. By rpX we shall denote the set X equipped with the /¿-uniform topology. We say that E and D subsets of X are //-separated when E is far from D in the proximity defined by p.By a base on a space X we mean a nest generated intersection ring (or equivalently, a strong delta normal base) in X [1 and 10]. It is known that each base D on X has associated a Wallman compactification W^X, V) and a Wallman realcompactification v(X, D). If pX is a uniform space, we shall denote by Z(pX) the base on rpX formed by uniform zero-sets (see [7 and 8]), v(pX) will denote the space v(X, Z(pX)) and ß{pX) the space W{X, Z{pX)). THEOREM 1. Let T be a space in which X is m-dense, let pY be a uniform space, and let f: X -► rpY be a continuous map. Then the following are equivalent:(a) / has a continuous extension f:T -> T~jpmY.

Extension of functions with small oscillation

“…The criteria for continuous extension on dense subspaces had been studied by several authors. (See, e.g., [1], [4]. )…”