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# complete uniform space

Let $X$ be a uniform space with uniformity $\mathcal{U}$. A filter $\mathcal{F}$ on $X$ is said to be a *Cauchy filter* if for each entourage $V$ in $\mathcal{U}$, there is an $F\in\mathcal{F}$ such that $F\times F\subseteq V$.

We say that $X$ is *complete* if every Cauchy filter is a convergent filter in the topology $T_{{\mathcal{U}}}$ induced by $\mathcal{U}$. $\mathcal{U}$ in this case is called a *complete uniformity*.

A *Cauchy sequence* $\{x_{i}\}$ in a uniform space $X$ is a sequence in $X$ whose section filter is a Cauchy filter. A Cauchy sequence is said to be convergent if its section filter is convergent. $X$ is said to be *sequentially complete* if every Cauchy sequence converges (every section filter of it converges).

Remark. This is a generalization of the concept of completeness in a metric space, as a metric space is a uniform space. As we see above, in the course of this generalization, two notions of completeness emerge: that of completeness and sequentially completeness. Clearly, completeness always imply sequentially completeness. In the context of a metric space, or a metrizable uniform space, the two notions are indistinguishable: sequentially completeness also implies completeness.

## Mathematics Subject Classification

54E15*no label found*

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