Extension by continuity in pointfree topology
type: Item
instance of: scholarly article
title: Extension by continuity in pointfree topology
zbMATH Open document ID: 0974.54007
DOI: 10.1023/A:1008673803010
author: S. S. Hong
author: Bernhard Banaschewski
published in: Applied Categorical Structures
publication date: 2001-12-10T00:00:00Z
review text: A common problem in topology is that of constructing a continuous extension \(\overline{f}: Z\to Y\) of a given continuous map \(f: X\to Y\) where \(X\) is a dense subspace of \(Z\). A well-known theorem due to Bourbaki establishes a criterion, in regular spaces, for the existence and uniqueness of such an extension in terms of the convergence of certain filters. In `pointfree' topology one still has a good notion of convergence for filters, due to the second author of this paper (a filter in a frame \(L\) is convergent iff it meets every cover of \(L\)); but for regular frames this condition is equivalent to containing the neighbourhood filter of a point, and hence it is easily shown to be inadequate to prove the corresponding theorem on extensions. In the present paper, the authors introduce a stronger notion of `preservation of covers', appropriate to the situation where one has a continuous map defined on a dense sublocale, and show that this suffices to prove the analogue of Bourbaki's extension theorem. The resulting theorem is very close in spirit to (the regular case of) the result commonly known as `Joyal's lemma', and indeed the latter occurs as a special case of it, as do several known reflection theorems in locale theory.
Mathematics Subject Classification ID: 54C20
Mathematics Subject Classification ID: 06D22
Mathematics Subject Classification ID: 03E25
Mathematics Subject Classification ID: 18B30
Mathematics Subject Classification ID: 54A20
zbMATH DE Number: 1563362
reviewed by: Peter Johnstone
MaRDI profile type: Publication
full work available at URL: a:1008673803010
OpenAlex ID: W2043248260
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description: scientific article; zbMATH DE number 1563362