**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 *T*_{0} 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 *d*^{op}-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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