An Elementary Extension of Tietze's Theorem

Abstract

Here C(X) and C(A) are the sets of continuous real-valued functions on X and A, respectively. F is an extension of f means F(x) =f(x) if x E A. While Theorem T is included in almost any text on point-set topology, none of the many books we surveyed mentions anything more general than Theorem T, either as a corollary or as an exercise. Of course, some texts observe that by translation one can extend a function f satisfying M1 <f(x) < M2, x E A, to a ftinction F satisfying M1 < F(x) < M2, x E X when M1 and M2 are any two constants, not just M2 = c = M as given in Theorem T. It should be observed that the original Tietze Theorem was stated for metric spaces and later generalized by Urysohn to normal Hausdorff spaces. Also, some extensions of Theorem T different from what is presented here appear in the literature (cf. [2]). This note considers the following question: Let A be a closed subset of X and assume g, h E C(X) such that g(x) < h(x) for every x E X. If f is a function in C(A) lying between g and h, i.e. g(x) <f(x) < h(x) for every x in A, can f be extended to a function F in C(X) lying between the same two functions? The following easy exercises lead directly from Theorem T to an affirmative answer to the question. However, we show later that there is a more general version that can be verified immediately from Theorem T.