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Apartness for continuous dcpos

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In constructive mathematics, we can distinguish between non-empty types and inhabited types. Similarly, we can distinguish between two elements being unequal and two elements being apart. Continuous dcpos are an important class of posets in theoretical computer science and topology. We introduce them by studying a particular example related to Dedekind reals. We then describe a notion of apartness for continuous dcpos and highlight its connections to both the Scott topology and the apartness topology of Bridges and Vîță.

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