Discrete Uniformity generates Discrete Topology
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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 $S$.
$\blacksquare$
Sources
- 1978: Lynn Arthur Steen and J. Arthur Seebach, Jr.: Counterexamples in Topology (2nd ed.) ... (previous) ... (next): Part $\text {II}$: Counterexamples: $1 \text { - } 3$. Discrete Topology: $11$