Enumerable and Dense

Question

I'm starting in topology! Something I'm not finding is the relation between enumerability and density, my question is, "A set being dense implies not being enumerable"?

Answer 1

Density does not imply that a set is enumerable or uncountable. A subset $E$ of a topological space $X$ is dense if every $x \in X$ is either in $E$ or is a limit point of $E$ (equivalently, if $\overline{E} = X$). The most common example you'll see of this is that $\mathbb{Q}$ (which happens to be enumerable) is dense in $\mathbb{R}$. However, $\mathbb{Q} \cup [0, 1]$ (with $[0, 1] \subset \mathbb{R}$) is dense in $\mathbb{R}$ but uncountable.