Pseudometric Space generates Uniformity

Theorem
Let $P = \struct {A, d}$ be a pseudometric space.

Let $\UU$ be the set of sets defined as:
 * $\UU := \set {u \in A \times A: \exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u}$

where:
 * $\R_{>0}$ is the set of strictly positive real numbers
 * $u_\epsilon$ is defined as:
 * $u_\epsilon := \set {\tuple {x, y}: \map d {x, y} < \epsilon}$

Then $\UU$ is a uniformity on $A$ which generates a uniform space with the same topology as the topology induced by $d$.

Proof
We check whether the Uniformity Axioms are satisfied.

$\text U 1$
Let $x \in A$.

Then $\tuple {x, x} \in \Delta_A$.

Let $u \in \UU$.

From definition:
 * $\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$

By :
 * $\map d {x, x} = 0 < \epsilon$

Hence we have $\tuple {x, x} \in u_\epsilon$.

By definition of subset, $\Delta_A \subseteq u_\epsilon \subseteq u$.

$\text U 2$
Let $u, v \in \UU$.

From definition:
 * $\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$
 * $\exists \delta \in \R_{>0}: u_\delta \subseteq v$

assume that $\delta \ge \epsilon$.

Let $\tuple {x, y} \in u_\epsilon$.

Then $\map d {x, y} < \epsilon \le \delta$.

Thus $\tuple {x, y} \in u_\delta$.

By definition of subset, $u_\epsilon \subseteq u_\delta$.

Hence:

$\text U 3$
Let $v \in A \times A$.

Suppose $u \in \UU$ and $u \subseteq v$.

From definition:
 * $\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$

Thus $u_\epsilon \subseteq v$.

Therefore $v \in \UU$ as well.

$\text U 4$
Let $u \in \UU$.

From definition:
 * $\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$

Let $v = u_{\epsilon / 2}$.

Since $u_{\epsilon / 2} \subseteq v$ and $\dfrac \epsilon 2 > 0$, $v \in \UU$.

From definition:

Let $\tuple {x, z} \in v \circ v$.

Then:
 * $\exists y \in A: \map d {x, y} < \dfrac \epsilon 2, \map d {y, z} < \dfrac \epsilon 2$

By :

Hence $\tuple {x, z} \in u_\epsilon \subseteq u$.

By definition of subset, $v \circ v \subseteq u$.

$\text U 5$
Let $u \in \UU$.

From definition:
 * $\exists \epsilon \in \R_{>0}: u_\epsilon \subseteq u$

Suppose $\tuple {b, a} \in u_\epsilon$.

Then $\map d {b, a} < \epsilon$.

By :
 * $\map d {a, b} = \map d {b, a} < \epsilon$

Hence $\tuple {a, b} \in u_\epsilon \subseteq u$.

Consider the set $u^{-1} = \set {\tuple {y, x}: \tuple {x, y} \in u}$.

From the above, $\tuple {b, a} \in u^{-1}$.

By definition of subset, $u_\epsilon \subseteq u^{-1}$.

Hence $u^{-1} \in \UU$.