Question

Prove or disprove every topological space is metrizible

Answer 1

First we prove that every finite metric space is discrete. Indeed, let $(X,\rho)$ be a metric space, where $X$ is finite.

Let $l(x)=\min\limits_{y\in X\setminus\{x\}}\rho(x,y)$ for every $x\in X$. Then $l(x)>0$, and so $B_{l(x)}(x)=\{x\}$ for every $x\in X$.

So we obtain that $(X,\rho)$ is discrete topological space.

But not every finite topological space is a discrete topological space, so not every topological space is metrizable.