# Definition of a Cauchy sequence in terms of a Cauchy filter

Prerequisites in case you may need and I am correct about them:

An entourage is a member of uniformity structure on a set for it to be a uniform space. Intuitively, an entourage is a relation on a set, such that it specifies those pairs of points, the "between-distance" for each pair being bounded by some "value" specified by the entourage. It is easier to understand for a metric space, which is an example of uniform space.

1. The definition I know for a Cauchy sequence in a uniform space is

A sequence $(x_i)$ is a Cauchy sequence if for every entourage $V$ there exists $n \in \mathbb{N}$ such that for all $i, j ≥ n$, $(x_i, x_j)$ is a member of $V$.

2. From Planetmath, the definition for a Cauchy sequence in a uniform space is

A Cauchy sequence $x_i$ in a uniform space $X$ is a sequence in $X$ whose section filter is a Cauchy filter,

where the section filter of a sequence is defined to be the maximal proper filter containing the filter base generated by the sequence, if I understand its definition correctly.

3. This definition of Cauchy sequence surprises me, because I was thinking instead

A Cauchy sequence $x_i$ in a uniform space $X$ is a sequence in $X$ whose filter is a Cauchy filter,

where the filter of a sequence is the minimal filter containing the filter base generated by the sequence.

So I wonder if the three definitions are equivalent?

Thanks and regards!

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I am not familiar with the notion of "entourage"; could you provide the definition? –  Arturo Magidin Feb 20 '12 at 18:16
@ArturoMagidin: An entourage is a member of uniformity structure on a set for it to be a uniform space. Wiki has the description here. It is easier to understand for a metric space, which is an example of uniform space. Intuitively, an entourage is a relation on a set, such that it specifies those pairs of points, the "distance" for each is within some "value". –  Tim Feb 20 '12 at 18:22
Please put relevant definitions in the body of the post; don't force people, like myself, who are unfamiliar with the term to wade into the comments to find out what the words mean. –  Arturo Magidin Feb 20 '12 at 18:26
@Arturo: For the record, I would expect anyone with a basic knowledge of uniform spaces to be familiar with the term entourage, just as I’d expect anyone with a basic knowledge of topological spaces to be familiar with the term neighborhood. –  Brian M. Scott Feb 20 '12 at 20:27
@Brian: Fair enough; I'm not familiar with either... –  Arturo Magidin Feb 20 '12 at 20:30

## 1 Answer

The equivalence of (1) and (3) is standard. The PlanetMath definition of section filter is simply wrong: it doesn’t define a unique object, since the filter generated by the sections (or tails, as I prefer to call them) may have many maximal extensions.

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