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Uniformly continuous maps send Cauchy sequences to Cauchy sequences
Let and be metric spaces and let be uniformly continuous.
Proposition: If is a Cauchy sequence in , then is a Cauchy sequence in .
This is an important structural feature: uniform continuity is exactly the hypothesis needed to transport completeness properties through a map.
A metric space is a pair where is a set and is a metric on .
Every metric space determines a topological space via the metric-induced topology, whose basic neighborhoods are open balls.
Let and . The function is uniformly continuous on if
Compared to ordinary continuity on , the key point is that depends only on , not on the location in .
Useful properties:
Examples in :
A Cauchy sequence in a metric space is a sequence such that for every there exists with
A complete metric space is a metric space in which every Cauchy sequence converges (as a convergent sequence) to a point of .