[[!redirects Cauchy sequence]] #Contents# * table of contents {:toc} ## 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 $T$ be a [[directed type]] and [[codirected type]] where the directed type operation $\oplus$ is associative, and let $S$ be a $T$-[[premetric space]]. A net $x: T \to S$ is a __$T$-Cauchy approximation__ if $$x:R_{+} \to S \vdash c(x):\prod_{\delta:R_{+}} \prod_{\eta:R_{+}} x_\delta \sim_{\delta \oplus \eta} x_\eta$$ Every $T$-Cauchy approximation is a [[Cauchy net]] indexed by $T$. This is because $T$ is a codirected type, with $N:T$ defined as $N \coloneqq \delta \otimes \eta$ for $\delta:T$ and $\eta:T$. $\epsilon:R_{+}$ is defined as $\epsilon \coloneqq \delta \oplus \eta$. ### In Cauchy spaces ### ... ### Cauchy sequences ### A __Cauchy sequence__ is a Cauchy net whose index type is the [[natural numbers]] $\mathbb{N}$. ## See also ## * [[Cauchy approximation]] * [[Cauchy structure]] * [[premetric space]] * [[net]] * [[filter]]