7.2 The Uniform Topology
Definition 7.2.1 (Uniform Topology).label Let $T$ be a set and $(X, \fU)$ be a uniform space. For each $U \in \fU$, let
then the uniform topology on $X^{T}$ is the topology induced by the uniformity generated by $\bracs{E(U)| U \in \fU}$.
Remark 7.2.1.label Properties such as completeness require the presence of a uniform structure. However, referring to it as the uniform uniformity is a little ridiculous. As such, in this section, completeness with respect to the uniform topology corresponds to completeness with respect to the uniform uniformity.
Proposition 7.2.2.label Let $X$ be a topological space and $Y$ be a uniform space, then:
- (1)
$C(X; Y) \subset Y^{X}$ is closed with respect to the uniform topology.
- (2)
If $X$ is a uniform space, then $UC(X; Y) \subset Y^{X}$ is closed with respect to the uniform topology.
In particular, if $Y$ is complete, then the above spaces are complete.
Proof. Let $x \in X$ and $V$ be an entourage of $Y$, then there exists a symmetric entourage $W$ of $Y$ such that $W \circ W \circ W \subset V$.
(1): Let $f \in \ol{C(X; Y)}$, then there exists $g \in C(X; Y)$ with $(f, g) \in E(W)$. Let $U \in \cn(x)$ such that $g(U) \subset W(g(x))$, then for any $y \in U$,
so $(f(x), f(y)) \in W \circ W \circ W \subset V$.
Therefore $f$ is continuous at $x$.
(2): Let $f \in \ol{UC(X; Y)}$, then there exists $g \in UC(X; Y)$ with $(f, g) \in E(W)$. Let $U$ be an entourage of $X$ such that $(g \times g)(U) \subset W$, then for any $(x, y) \in U$,
so $(f(x), f(y)) \in W \circ W \circ W \subset V$.
If $Y$ is complete, then $Y^{T}$ with the uniform topology is complete by Proposition 7.1.5. Thus $C(T; X)$ and $UC(T; X)$ are both complete subspaces by Proposition 6.5.3.$\square$