nLab
Scott topology

Contents

Definition

The Scott topology on a preordered set is the topology in which the open subsets (called Scott-open) are precisely those whose characteristic functions (from the given preorder into the preorder of truth values) preserve directed joins (and this makes them necessarily monotonic).

This in fact ensures that, in general, the continuous functions between preorders with Scott topologies are precisely those (necessarily monotonic) functions between them which preserve directed joins (called Scott-continuous). The poset of truth values itself, therefore, when equipped with the Scott topology, becomes the open-set classifier, Sierpinski space.

Properties

As injective objects in T 0T_0-spaces

In the category of T 0T_0 topological spaces (see at separation axiom), the injective objects are precisely those given by Scott topologies on continuous lattices; as locales these are locally compact and spatial.

Revised on March 17, 2016 17:12:24 by Urs Schreiber (194.210.159.117)