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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constructive mathematics, realizability, computability
propositions as types, proofs as programs, computational trinitarianism
The Cauchy completion of the rational numbers, in the sense that all Cauchy nets or Cauchy filters in the rational numbers converge.
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Let be the rational numbers and let
be the set of positive rational numbers. Let be a directed set and
is the set of all Cauchy nets with index set and values in . Let
be the (large) set of all Cauchy nets in in a universe , where is the category of all directed sets in .
Let the relation in the Cartesian product for directed sets and be defined as
Let a generalized Cauchy algebra be defined as a set with a function such that
A generalized Cauchy algebra homomorphism is a function between Cauchy algebras and such that
The category of generalized Cauchy algebras is the category whose objects are generalized Cauchy algebras and whose morphisms are generalized Cauchy algebra homomorphisms. The set of generalized Cauchy real numbers, denoted , is defined as the initial object in the category of Cauchy algebras.
Last revised on May 24, 2023 at 13:36:47. See the history of this page for a list of all contributions to it.