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Apartness and Uniformity - A Constructive Development (Paperback, 2011)
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Apartness and Uniformity - A Constructive Development (Paperback, 2011)
Series: Theory and Applications of Computability
Expected to ship within 10 - 15 working days
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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.
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