Properly Injective Spaces and Function Spaces (1997)
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BibTeX
@MISC{Escardo97properlyinjective,
author = {Martín Hötzel Escardo},
title = {Properly Injective Spaces and Function Spaces},
year = {1997}
}
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Abstract
Given an injective space D (a continuous lattice endowed with the Scott topology) and a subspace embedding j : X ! Y , Dana Scott asked whether the higher-order function [X ! D] ! [Y ! D] which takes a continuous map f : X ! D to its greatest continuous extension ¯ f : Y ! D along j is Scott continuous. In this case the extension map is a subspace embedding. We show that the extension map is Scott continuous iff D is the trivial one-point space or j is a proper map in the sense of Hofmann and Lawson. In order to avoid the ambiguous expression "proper subspace embedding", we refer to proper maps as finitary maps. We show that the finitary sober subspaces of the injective spaces are exactly the stably locally compact spaces. Moreover, the injective spaces over finitary embeddings are the algebras of the upper power space monad on the category of sober spaces. These coincide with the retracts of upper power spaces of sober spaces. In the full subcategory of locally compact sober spaces, t...
