nLab modulus of convergence

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Definition

Let \mathbb{Q} be the rational numbers and let

+{x|0<x}\mathbb{Q}_{+} \coloneqq \{x \in \mathbb{Q} \vert 0 \lt x\}

be the positive rational numbers. Let SS be a Booij premetric space, and let x:ISx:I \to S be a net with index set II. A II-modulus of convergence is a function M +IM \in {\mathbb{Q}_{+}} \to I such that for all positive rational numbers ϵ\epsilon and all indices iIi \in I and jIj \in I, if iM(ϵ)i \geq M(\epsilon) and jM(ϵ)j \geq M(\epsilon), then x i ϵx jx_i \sim_{\epsilon} x_j.

The composition xMx \circ M of a net xx and a modulus of convergence MM is also a net.

See also

References

Last revised on May 31, 2022 at 13:29:24. See the history of this page for a list of all contributions to it.