Is there a definition Df(g) of uniform continuity of g, without using the notion of metric?


Let $(E,d\_E)$ and $(F, d\_F)$ metrics spaces, $f$ continuous fonction of $E$ to $F$


We must have :


Df$(f)$ iff $f$ is uniform continue.



About uniform continuity
The notion of uniform continuity does not require all of the structure behind metric spaces. Every metric space is automatically a uniform space, and the uniformly continuous functions are the morphisms in the category of uniform spaces.


The two most common ways of defining uniform spaces are in terms of entourages or also in terms of uniform covers. The entourage definition of a uniform space is simpler while the uniform cover definition of a uniform space generalizes better to point-free topology and gives some very good motivation to the notion of paracompactness.


We shall now give the entourage definition of a uniformity. Let $X$ be a set. Then a uniformity is a filter $\mathcal{U}$ on the set $X\times X$ that satisfies the following properties:


1. If $E\in\mathcal{U}$, then $\{(x,x)\mid x\in X\}\subseteq E$.
2. If $E\in\mathcal{U}$, then $E^{-1}\in\mathcal{U}$.
3. If $E\in\mathcal{U}$, then there exists some $F\in\mathcal{U}$ where
$F\circ F\subseteq E$.


If $\bigcap\mathcal{F}=\{(x,x)\mid x\in X\}$, then we say that the uniformity $\mathcal{F}$ is separated.


Every metric space $(X,d)$ is endowed with a uniformity $\mathcal{F}\_{d}$ such that
$E\in\mathcal{F}\_{d}$ if and only if there exists an $\epsilon>0$ where
$\{(x,y)\mid d(x,y)<\epsilon\}\subseteq E$.


The $\epsilon-\delta$-definition of uniform continuity generalizes to uniform spaces quite easily. If $(X,\mathcal{F}),(Y,\mathcal{G})$ are uniform spaces, then we say that a mapping $f:X\rightarrow Y$ is uniformly continuous if for all $E\in\mathcal{G}$, there exists some $D\in\mathcal{F}$ such that if $(x,y)\in D$, then $(f(x),f(y))\in E$.


**More facts about uniform spaces**


Each uniform space $(X,\mathcal{F})$ is automatically endowed with a topology where a set $U$ is open if and only if for each $x\in U$, there exists an entourage $E$ where $E[x]\subseteq U$. Furthermore, every separated uniform space is automatically completely regular in this topology.


Every completely regular space can be endowed with a compatible uniformity.


Every compact Hausdorff space has a unique compatible uniformity, and if $C$ is a compact space, and $X$ is a uniform space, then every continuous mapping
$f:C\rightarrow X$ is uniformly continuous.


Topological groups can also be endowed with a three compatible uniformities, namely the left uniformity, the right uniformity, and the intersection of these two uniformities.


**References**


General topology textbooks (such as Willard Ch 9, and Kelley as was observed by Willie Wong or by Bourbaki Chapter II as noted by abx) often have a chapter on uniform spaces. There are a couple good general topology texts on uniform spaces including the textbook by John Isbell (which uses the uniform cover definition of uniformity) as well as the book by I.M. James (which uses the entourage definition of a uniformity).


