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.
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