## Definition ## Let $T$ be a [[directed type]], let $S$ be a $T$-[[premetric space]], and let $x:I \to S$ be a [[net]] with index type $I$. A __$T$-modulus of Cauchy convergence__ is a function $M: T \to I$ with a type $$\mu:\prod_{\epsilon:T} \prod_{i:I} \prod_{j:I} (i \geq M(\epsilon)) \times (j \geq M(\epsilon)) \to (x_i \sim_{\epsilon} x_j)$$ The composition $x \circ M$ of a net $x$ and a $T$-modulus of Cauchy convergence $M$ is also a [[net]]. ## See also ## * [[limit of a net]] * [[premetric space]] * [[Cauchy structure]] * [[Cauchy approximation]] * [[HoTT book real numbers]] ## References ## * Auke B. Booij, Analysis in univalent type theory ([pdf](https://etheses.bham.ac.uk/id/eprint/10411/7/Booij2020PhD.pdf)) * Univalent Foundations Project, [[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)