28 ERRETT A. BISHOP constructivization of algebra would seem hardly to have begun. The problems are formidable. A very tentative suggestion is that we should restrict our attentions to algebraic structures endowed with some sort of topology, with respect to which all operations and maps are continuous. The work of Tennenbaum quoted above might provide some ideas of how to accomplish this. The task is complicated by the circumstance that no completely suitable con- structive framework for general topology has yet been found. The constructivization of general topology is impeded by two obstacles. First, the classical notion of a topological space is not constructively viable. Second, even for metric spaces the classical notion of a continuous function is not constructively viable the reason is that there is no constructive proof that a (pointwise) continuous function from a compact (complete and totally bounded) metric space to IR is uniformly continuous. Since uniform continuity for functions on a compact space is the useful concept, pointwise continuity (no longer useful for proving uniform continuity) is left with no useful function to perform. Since uniform continuity cannot be formulated in the context of a general topological space, the latter concept also is left with no useful func- tion to perform. In [ 11 I was able to get along by working mostly with metric spaces and using various ad hoc definitions of continuity: one for compact spaces, anoth- e r for locally compact spaces, and another for the duals of Banach spaces. The unpublished manuscript [4] was an attempt to develop constructive general topology systematically. The basic idea is that a topological space should con- sist of a set X, endowed with both a family of metrics and a family of bound- edness notions, where a boundedness notion on X is a family S of subsets of X (called bounded subsets), whose union is X, closed under finite unions and the formation of subsets. For example, let C be the set of all real ~ l u e d functions f: IR + W , bouncied and (uniformly) continuous on finite intervals. Each finite interval of iR induces a metric on C (the uniform metric on that interval). In addi- tion, there is a natural boundedness notion S. A subset E of C belongs to S if there exists r 0 such athat If ( s r for all f in E. A sequence {fn 1 of elements of C converges to an element f of C if it converges with re- spect to each of the metrics on C, and if it is bounded. The notion of a continuous function from one such space to another, as given in 141, is somewhat involved and will not be repeated here. It was
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