The theory presented in this book is developed constructively, is based
on a few axioms encapsulating the notion of objects (points and sets)
being apart, and encompasses both point-set topology and the theory of
uniform spaces. While the classical-logic-based theory of proximity
spaces provides some guidance for the theory of apartness, the notion of
nearness/proximity does not embody enough algorithmic information for a
deep constructive development. The use of constructive (intuitionistic)
logic in this book requires much more technical ingenuity than one finds
in classical proximity theory -- algorithmic information does not come
cheaply -- but it often reveals distinctions that are rendered invisible
by classical logic.
In the first chapter the authors outline informal constructive logic and
set theory, and, briefly, the basic notions and notations for metric and
topological spaces. In the second they introduce axioms for a point-set
apartness and then explore some of the consequences of those axioms. In
particular, they examine a natural topology associated with an apartness
space, and relations between various types of continuity of mappings. In
the third chapter the authors extend the notion of point-set
(pre-)apartness axiomatically to one of (pre-)apartness between subsets
of an inhabited set. They then provide axioms for a quasiuniform space,
perhaps the most important type of set-set apartness space. Quasiuniform
spaces play a major role in the remainder of the chapter, which covers
such topics as the connection between uniform and strong continuity
(arguably the most technically difficult part of the book), apartness
and convergence in function spaces, types of completeness, and neat
compactness. Each chapter has a Notes section, in which are found
comments on the definitions, results, and proofs, as well as occasional
pointers to future work. The book ends with a Postlude that refers to
other constructive approaches to topology, with emphasis on the relation
between apartness spaces and formal topology.
Largely an exposition of the authors' own research, this is the first
book dealing with the apartness approach to constructive topology, and
is a valuable addition to the literature on constructive mathematics and
on topology in computer science. It is aimed at graduate students and
advanced researchers in theoretical computer science, mathematics, and
logic who are interested in constructive/algorithmic aspects of
topology.
