Scott continuity

In mathematics, given two partially ordered sets P and Q, a function ${\displaystyle f:P\rightarrow Q}$ between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema, i.e. if for every directed subset D of P with supremum in P its image has a supremum in Q, and that supremum is the image of the supremum of D: that is, ${\displaystyle \sqcup f[D]=f(\sqcup D)}$, where ${\displaystyle \sqcup }$ is the directed join. When ${\displaystyle Q}$ is the poset of truth values, i.e. Sierpinski space, then the ${\displaystyle f}$ are characteristic functions, and thus, Sierpinski space is the classifying topos for open sets.

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