Indiscrete Space is Pseudometrizable

Theorem
Let $T = \left({S, \left\{{\varnothing, S}\right\}}\right)$ be an indiscrete topological space.

Then $T$ is pseudometrizable.

Proof
Let $d: S\times S \to \mathbb{R}$ be a function defined by setting:


 * $d(x,y)=0 \ \forall x,y\in S$.

Then clearly $d$ is a pseudometric.

Let $\left({S, \vartheta_{\left({S, d}\right)}}\right)$ be the topological space induced by $d$.

Since $\left({S,\vartheta_{\left({S,d}\right)}}\right)$ is a topological space, by Empty Set is Element of Topology we find that $\varnothing$ is open.

Clearly, for any $\epsilon$ we have:


 * $N_\epsilon \left({a}\right) := \left\{{x \in S: d \left({x, a}\right) < \epsilon}\right\} = S$.

Here $N_\epsilon \left({a}\right)$ is the neighborhood of $a$.

Let $U\subset S$ be a non-empty open set. By Open Sets in Pseudometric Space we see that for every $x\in U$, there must exist an $\epsilon > 0$ such that $N_\epsilon \left({x}\right)\subset U$.

However, $S\subset N_\epsilon \left({x}\right)$.

Thus $U=S$.

Clearly $\varnothing$ and $S$ are the only open sets of this pseudometric space.