Scott topology

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.

Revised on February 24, 2015 09:59:34 by Noam Zeilberger (