Homotopy Type Theory limit of a net > history (Rev #6, changes)

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Definition
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Definition

In premetric spaces

Let $T$ be a directed type, and let $S$ be a $T$ -premetric space. Given a directed type $I$ , a limit of a net $x: I \to S$ is a term $l:S$ with

\lambda: \prod_{\epsilon:T} \Vert \sum_{N:I} \prod_{i:I} (i \geq N) \to (x_i \sim_{\epsilon} l) \Vert

Cauchy approximations

Let $A$ be a dense Archimedean ordered abelian group with a point $1:A$ and a term $\zeta: 0 \lt 1$ . Let $A_{+} \coloneqq \sum_{a:A} (0 \lt a)$ be the positive cone? of $A$ .

A limit of a $A_{+}$ -Cauchy approximation $x: A_{+} \to S$ is a term $l:S$ with

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

In convergence spaces

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Sequences

A limit of a sequence is a limit of a net that happens to be a sequence.