# uniform continuity

In this entry, we extend the usual definition of a uniformly continuous function between metric spaces to arbitrary uniform spaces.

Let $(X,\mathcal{U}),(Y,\mathcal{V})$ be uniform spaces (the second component is the uniformity on the first component). A function $f:X\to Y$ is said to be *uniformly continuous* if for any $V\in \mathcal{V}$ there is a $U\in \mathcal{U}$ such that for all $x\in X$, $U[x]\subseteq {f}^{-1}(V[f(x)])$.

Sometimes it is useful to use an alternative but equivalent^{} version of uniform continuity of a function:

###### Proposition 1.

Suppose $f\mathrm{:}X\mathrm{\to}Y$ is a function and $g\mathrm{:}X\mathrm{\times}X\mathrm{\to}Y\mathrm{\times}Y$ is defined by $g\mathit{}\mathrm{(}{x}_{\mathrm{1}}\mathrm{,}{x}_{\mathrm{2}}\mathrm{)}\mathrm{=}\mathrm{(}f\mathit{}\mathrm{(}{x}_{\mathrm{1}}\mathrm{)}\mathrm{,}f\mathit{}\mathrm{(}{x}_{\mathrm{2}}\mathrm{)}\mathrm{)}$. Then $f$ is uniformly continuous iff for any $V\mathrm{\in}\mathrm{V}$, there is a $U\mathrm{\in}\mathrm{U}$ such that $U\mathrm{\subseteq}{g}^{\mathrm{-}\mathrm{1}}\mathit{}\mathrm{(}V\mathrm{)}$.

###### Proof.

Suppose $f$ is uniformly continuous. Pick any $V\in \mathcal{V}$. Then $U\in \mathcal{U}$ exists with $U[x]\subseteq {f}^{-1}(V[f(x)])$ for all $x\in X$. If $(a,b)\in U$, then $b\in U[a]\subseteq {f}^{-1}(V[f(a)])$, or $f(b)\subseteq V[f(a)]$, or $g(a,b)=(f(a),f(b))\in V$. The converse^{} is straightforward.
∎

Remark. Note that we could have picked $U$ so the inclusion becomes an equality.

###### Proposition 2.

. If $f\mathrm{:}X\mathrm{\to}Y$ is uniformly continuous, then it is continuous^{} under the uniform topologies of $X$ and $Y$.

###### Proof.

Let $A$ be open in $Y$ and set $B={f}^{-1}(A)$. Pick any $x\in B$. Then $y=f(x)$ has a uniform neighborhood $V[y]\subseteq A$. By the uniform continuity of $f$, there is an entourage $U\in \mathcal{U}$ with $x\in U[x]\subseteq {f}^{-1}(V[y])\subseteq {f}^{-1}(A)=B$. ∎

Remark. The converse is not true, even in metric spaces.

Title | uniform continuity |
---|---|

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 |