Discrete Uniformity generates Discrete Topology

Theorem
Let $S$ be a set.

Let $\UU$ be the discrete uniformity on $S$.

Then the topology generated by $\UU$ is the discrete topology on $S$.

The diagonal relation $\Delta_S$ generates the basis for this discrete topology.

Proof
From the construction, let $\tau \subseteq \powerset S$ be a subset of the power set of $S$, created from $\UU$ by:


 * $\tau := \set {\map u x: u \in \UU, x \in X}$

where:
 * $\forall x \in X: \map u x = \set {y: \tuple {x, y} \in u}$

We need to show that $\tau$ is the discrete topology.

Consider $\Delta_S \in \UU$.

Let $x \in S$.

Then the set:
 * $U_x := \set {y: \tuple {x, y} \in \Delta_S}$

We have that: $\forall x \in S: U_x = \set x \in \UU$

Thus we have:
 * $\BB := \set {U_x: x \in S} = \set {\set x: x \in S}$

From Basis for Discrete Topology, we have that $\BB$ is a basis for the discrete topology on $S$.