Continuity properties in constructive mathematics

The purpose of this paper is an axiomatic study of the interrelations between certain continuity properties. We deal with principles which are equivalent to the statements "every mapping is sequentially nondiscontinuous", "every sequentially nondiscontinuous mapping is sequentially continuous", and "every sequentially continuous mapping is continuous". As corollaries, we show that every mapping of a complete separable space is continuous in constructive recursive mathematics (the Kreisel-LacombeSchoenfield-Tsejtin theorem) and in intuitionism. As early as 1954 Markov had obtained the first continuity result in constructive recursive mathematics that every mapping from R to R is sequentially nondiscontinuous. (We say that a mapping between metric spaces is sequentially nondiscontinuous if xn -+ x as n -+ o and d(f(xn), f(x)) ? 6 for all n imply 6 0 there exists 6 > 0 such that d(x, y) < 6 implies d(f(x), f(y)) < E for all y E X.) For applications of these results to concrete mathematical problems, see [2, Chapter XVI]. Beeson [1] carefully studied the KLST theorem, and showed that Markov's principle MP implies KLST, but the converse is unprovable, in the intuitionistic formal system of Heyting's arithmetic HA with Church's thesis CT. (Constructive recursive mathematics could be formalized relative to the system HA with CT and MP.) On the other hand, Troelstra [12] proved results corresponding to the KLST theorem and to Orevkov's theorem in intuitionism, which is inconsistent with CT. Received February 5, 1991; revised May 3, 1991. 1991 Mathematics Subject Classification. Primary 03F65; Secondary 46S30.