3 CONSTRUCTIVE CONTINUITY 1.3

Finally, there is the constructive approach to analysis initiated by

Bishop [B]? this is the approach followed here. Based only on our innate

conceptions of integer and of finite procedure, and unencumbered by any

formal logical system, this approach uses an unrestrictive notion of real

number and an unformalized concept of function. The approach of [B] is

described thoroughly in [S]. The continuity problems are discussed on

page 70 of [B],where it is predicted that they will never be solved.

Thus we follow here a direct mathematical, rather than

metamathematical, approach, using no formal system, under no special

hypotheses, and with no preconceived notions concerning the nature of the

continuum, other than that it consists of constructively defined Cauchy

sequences of rational numbers.

1.3. Reliefs. In this paper we introduce the method of reliefs for

the study of continuity. This consists of a constructivization of the

method of Urysohn's Lemma (which characterizes normal topological spaces

in terms of the separation of closed sets by continuous functions).

Urysohn's method involves the construction of a family {X }„6D of

subsets of a space X, indexed by a dense set D of real numbers, and

ordered so that X c x„ when a 0. A real-valued function is defined

a 0

on X by

f(x) s inf {a e D : x e X^} (x e X)

When, in addition, the sets X are open, and X c x„ whenever a 0,

a a 0

then the generated function is continuous (see pp. 146-9 of [D],pp. 43-4

of [GJ],or pp. 113-5 of [K]).

Naturally, more effort is required to obtain a constructively defined

function. The indicated infima do not always exist constructively, and

the classical continuity proof is based on nonconstructive methods.

A family of subsets of a metric space X, ordered by a dense set of

real numbers, will be called a relief of X. Our first task is to find

conditions on a relief under which the infima indicated above do exist

constructively. The key condition for this turns out surprisingly simple.