[FOM] the role of local compactness

Stephen G Simpson simpson at math.psu.edu
Thu Mar 23 14:50:38 EST 2006

Recent postings by Bauer and Waaldijk point up a difficulty in the
Bishop-style treatment of continuous functions.  Namely, Bishop's
definition works well only in locally compact spaces, e.g.,
n-dimensional Euclidean space, R^n.

I want to remark that, in my book "Subsystems of Second Order
Arithmetic" (Springer, 1999, http://www.math.psu.edu/simpson/sosoa/),
I use a different approach, which assumes neither local compactness
nor uniform continuity on compact subspaces.  My approach works well
even on non-locally-compact spaces such as the Baire space, N^N, and
infinite dimensional Euclidean space, R^N.  For example, I prove a
version of the Tietze Extension Theorem in this kind of context.  One
should perhaps compare my results to those mentioned by Bauer and

A couple of footnotes:

1. In contrast to Bauer and Waaldijk, my book is not about
constructivism or intuitionism.  Instead, I deal with subsystems of
classical second-order arithmetic.  The subsystems vary greatly in
proof-theoretic strength, from very strong (Pi^1_n comprehension, n =
1,2,3,...) to very weak (PRA or weaker).  Throughout the book I assume
the law of the excluded middle.

2. The discussion of topology in my book is restricted to complete
separable metric spaces.  However, in recent work, Carl Mummert and I
extend the discussion to a much wider class of topological spaces,
including many which are not metrizable.  See our paper in the
Bulletin of Symbolic Logic, 11, 2005, pp. 526-533.  And, in still more
recent work by Mummert and Frank Stephan, it is shown that that the
class of spaces which Mummert and I dealt with can be characterized
precisely as the second-countable T_1 spaces with the strong Choquet

Name: Stephen G. Simpson

Affiliation: Professor of Mathematics, Pennsylvania State University

Research interests: mathematical logic, foundations of mathematics

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