Proof or counterexample related topology of metric spaces

Question

Let $(E,m)$ be a metric space with the topology induced by the metric. I want to prove or disprove this

If no open ball is contractible then $E$ has caridinality as much countable and all its points all point of accumulation.

Answer 1

As a simple counterexample, let $E=\,\mathbb{R}{\setminus}\mathbb{Q}$, with the metric inherited from $\mathbb{R}$.

Then $E$ is uncountable, and no open ball is contractible.