nearness preserving maps extension of (uniformly) continluou s maps J Every uniformly continuous function from a dense subspace of a unifom space into a complete uniform space has a u.ni:~ormly continuou,s extension. This well-known theorem ka:: no direct topological counterpart. The reason becomes obvious in the realm of nearness structures: the concept of a subspace is adequate for uniform spaces but not car topological spaces.
In this paper it will be shown that every nearness preserving function from a dense subspace of a nearness space into a complete, regular nearness space has a nearness p;*eserving extension. An np#ication to uniform spaces yields the above-mentioned result. An application to topological spaces yields a necessary and sufficient condition for the extendibility of continuous func:tWs [ 21. The importance of nearxbess structures fC;r investiga:i'ions concerning the extendibility of continuous function has been first observed by Naimpally i4'4, the clue being that for any topological space (X, r) and arly subset S of X the nearness structure induced on S by r is usually much. "rkher" thaln the topological structure induced on S by 7.
For a definition of the concepts concerning nearness structures, the reader is referred to [ 3 1.