nLab modulated Cauchy complete space

Idea

A notion of Cauchy space where only certain Cauchy nets with a particular modulus of convergence converges in the space.

Definition

Let $S$ be a Cauchy space (such as a uniform space or a metric space). Given a directed set $I$, $S$ is $I$-modulated Cauchy complete if every Cauchy net in $S$ with index set $I$ and with a $I$-modulus of convergence converges.

A Cauchy space is said to be sequentially modulated Cauchy complete if every Cauchy sequence in $S$ with a $\mathbb{N}$-modulus of convergence converges.

Examples

• The Dedekind real numbers are sequentially modulated Cauchy complete.
• The HoTT book real numbers are sequentially modulated Cauchy complete.

See also

• Cauchy space
• complete space
• sequentially Cauchy complete space