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From Sets and Types to Topology and AnalysisTowards practicable foundations for constructive mathematics$
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Laura Crosilla and Peter Schuster

Print publication date: 2005

Print ISBN-13: 9780198566519

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198566519.001.0001

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From Sets and Types to Topology and Analysis

A. Bucalo

G. Rosolini

Oxford University Press

The constructive approach to topology has produced a wealth of new insights about spaces, almost reaching the point of making these more elementary than sets. In particular, the point of view taken by the notion of the basic picture as developed in the context of formal topology brings forward the relevance of relations between sets as an elementary constituent for understanding topological spaces. The chapter employs the language and notions of category theory to put the concept of the basic pair under scrutiny. It gives a presentation of topological spaces, which exploits another monoidal structure on the category of complete sup-lattices compared with the one given by Galois connections as in the memoirs by Joyal and Tierney. In this modified context, this chapter characterizes the frames which are topologies.

Keywords:   formal topology, basic picture, relations, topological space, category theory, Joyal, Tierney

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