uniform continuity
In this entry, we extend the usual definition of a uniformly continuous function between metric spaces to arbitrary uniform spaces.
Let be uniform spaces (the second component is the uniformity on the first component). A function is said to be uniformly continuous if for any there is a such that for all , .
Sometimes it is useful to use an alternative but equivalent version of uniform continuity of a function:
Proposition 1.
Suppose is a function and is defined by . Then is uniformly continuous iff for any , there is a such that .
Proof.
Suppose is uniformly continuous. Pick any . Then exists with for all . If , then , or , or . The converse is straightforward. ∎
Remark. Note that we could have picked so the inclusion becomes an equality.
Proposition 2.
. If is uniformly continuous, then it is continuous under the uniform topologies of and .
Proof.
Let be open in and set . Pick any . Then has a uniform neighborhood . By the uniform continuity of , there is an entourage with . ∎
Remark. The converse is not true, even in metric spaces.
Title | uniform continuity |
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Canonical name | UniformContinuity |
Date of creation | 2013-03-22 16:43:15 |
Last modified on | 2013-03-22 16:43:15 |
Owner | CWoo (3771) |
Last modified by | CWoo (3771) |
Numerical id | 7 |
Author | CWoo (3771) |
Entry type | Definition |
Classification | msc 54E15 |
Related topic | UniformlyContinuous |
Related topic | UniformContinuityOverLocallyCompactQuantumGroupoids |
Defines | uniformly continuous |