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Non-Hausdorff Topology and Domain Theory by Jean Goubault-Larrecq

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7.4 A weak form of completeness: Yoneda-completeness

Another notion of completeness was proposed by Bonsangue et al. (1998). This has come to be known in the literature as Yoneda-completeness. We shall see that this notion of completeness also has a natural characterization in terms of formal balls.

Definition 7.4.1 (Yoneda-complete) A hemi-metric space X, d is Yoneda-complete if and only if d is T0 and every Cauchy net in X, d has a d-limit.

The definition of Smyth-complete spaces was the same, except that we required every Cauchy net to have a dop-limit, not a d-limit.

Lemma 7.4.2 (Smyth ⇒ Yoneda-complete) Every Smyth-complete quasi-metric space is Yoneda-complete.

Proof In a Smyth-complete quasi-metric space, every Cauchy net has a limit in ...

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