Approach Spaces: The Missing Link In The Topology-uniformity-metric Triad (oxford Mathematical Monographs)
by R. Lowen /
1997 / English / PDF
7.9 MB Download
In topology the three basic concepts of metrics, topologies and
uniformities have been treated so far as separate entities by means
of different methods and terminology. This is the first book to
treat all three as a special case of the concept of approach
spaces. This theory provides an answer to natural questions in the
interplay between topological and metric spaces by introducing a
uniquely well suited supercategory of TOP and MET. The theory makes
it possible to equip initial structures of metricizable topological
spaces with a canonical structure, preserving the numerical
information of the metrics. It provides a solid basis for
approximation theory, turning ad hoc notions into canonical
concepts, and it unifies topological and metric notions. The book
explains the richness of approach structures in great detail; it
provides a comprehensive explanation of the categorical set-up,
develops the basic theory and provides many examples, displaying
links with various areas of mathematics such as approximation
theory, probability theory, analysis and hyperspace theory.
In topology the three basic concepts of metrics, topologies and
uniformities have been treated so far as separate entities by means
of different methods and terminology. This is the first book to
treat all three as a special case of the concept of approach
spaces. This theory provides an answer to natural questions in the
interplay between topological and metric spaces by introducing a
uniquely well suited supercategory of TOP and MET. The theory makes
it possible to equip initial structures of metricizable topological
spaces with a canonical structure, preserving the numerical
information of the metrics. It provides a solid basis for
approximation theory, turning ad hoc notions into canonical
concepts, and it unifies topological and metric notions. The book
explains the richness of approach structures in great detail; it
provides a comprehensive explanation of the categorical set-up,
develops the basic theory and provides many examples, displaying
links with various areas of mathematics such as approximation
theory, probability theory, analysis and hyperspace theory.
