TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 324. Number I. March 1991 EXTENDING DISCRETE-VALUED FUNCTIONS JOHN KULESZA, RONNIE LEVY, AND PETER NYIKOS ABSTRACT. In this paper, we show that for a separable metric space X, every continuous function from a subset S of X into a finite discrete space extends to a continuous function on X if and only if every continuous function from S into any discrete space extends to a continuous function on X. We also show that if there is no inner model having a measurable cardinal, then there is a metric space X with a subspace S such that every 2-valued continuous function from S extends to a continuous function on all of X, but not ev- ery discrete-valued continuous function on S extends to such a map on X. Furthermore, if Martin's Axiom is assumed, such a space can be constructed so that not even co-valued functions on S need extend. This last result uses a version of the Isbell-Mrowka space 'P having a C· -embedded infinite discrete subset. On the other hand, assuming the Product Measure Extension Axiom, no such 'P exists. O. INTRODUCTION Suppose that X is a metric space and S is a subset of X . Then the following statements are equivalent: (i) Every continuous function f: S -+ lR extends to a continuous function F: X -+ lR, and (ii) Every continuous function f: S -+ lR having a compact image extends to a continuous function F: X -+ lR. (Each statement is equivalent to S being a closed subset of X. To see this, notice that (i) trivially implies (ii), and every closed set satisfies (i) by Tietze's Exten- sion Theorem. Finally, if S is not closed in X, there is a sequence (] in S converging to a point p not in X. Then any function f: (] -+ {O, I} such that each fiber is infinite extends to a continuous F: S -+ [0, I]. If F is not onto, it can be composed with a continuous function from the range of F onto {O, I}. In any case, we get a continuous function from S onto a compact set which does not extend continuously to p.) In this paper we deal with the question of what happens when lR is replaced by a discrete space. In particular, we discuss the following problem: Suppose that X is a metric space and S is a subset such that every continuous function f: S -+ {O, I} extends to a continuous function F:X -+ {O, I}. Does it follow that every continuous function from S to a discrete space extends to a continuous function defined on all of X? We show that if X is separable, then the answer is "yes". Received by the editors September 30, 1988 and, in revised form, March 14, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 54C20, 54E35. Key words and phrases. 'P, PMEA, a-embedding. 293 © 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use