Homotopy Type Theory Cauchy net > history (Rev #5)

Definition
See also

Definition

In premetric spaces

Let $T$ be a directed type, and let $S$ be a $T$ -premetric space. Given a directed type $I$ , a net $x: I \to S$ is a Cauchy net if

x:I \to S \vdash c(x):\prod_{\epsilon:T} \Vert \sum_{N:I} \prod_{i:I} \prod_{j:I} (i \geq N) \times (j \geq N) \times (x_i \sim_{\epsilon} x_j) \Vert

Cauchy approximations

Let $R$ be a Archimedean ordered integral domain with a dense strict order, and let $R_{+}$ be the semiring? of positive terms in $R$ . If both $T$ and $I$ are $R_{+}$ , then a net $x: R_{+} \to S$ is a Cauchy approximation if

x:R_{+} \to S \vdash c(x):\prod_{\delta:R_{+}} \prod_{\eta:R_{+}} x_\delta \sim_{\delta + \eta} x_\eta

In Cauchy spaces

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Cauchy sequences

A Cauchy sequence is a Cauchy net whose index type is the natural numbers $\mathbb{N}$ .