analysis (differential/integral calculus, functional analysis, topology)
metric space, normed vector space
open ball, open subset, neighbourhood
convergence, limit of a sequence
compactness, sequential compactness
continuous metric space valued function on compact metric space is uniformly continuous
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Let $\mathbb{Q}$ be the rational numbers and let
be the set of positive rational numbers. Let $(S, \sim)$ be a Booij premetric space. A Cauchy approximation is a function $x: \mathbb{Q}_{+} \to S$ such that for all positive rational numbers $\delta$ and $\eta$, $x(\delta) \sim_{\delta + \eta} x(\eta)$.
The set of all Cauchy approximations is defined as
Every Cauchy approximation is a Cauchy net indexed by the positive rational numbers $\mathbb{Q}_{+}$.
A Cauchy approximation is the composition $x \circ M$ of a net $x$ and an modulus of convergence $M$.
Auke B. Booij, Analysis in univalent type theory (pdf)
Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)
Last revised on May 31, 2022 at 13:28:46. See the history of this page for a list of all contributions to it.