Separable space

Definition

A topological space $X$ is separable when there is a countable subset $D \subseteq X$ that is dense in $X$.

Remarks

The real line is separable, because $\mathbb{Q}$ is countable and dense. In the same way $\mathbb{R}^n$ is separable, with the dense set $\mathbb{Q}^n$. Each second-countable space is separable. For metric spaces, the converse is also true. An uncountable set with the discrete metric is not separable, because each dense subset is the whole set.

Explanations

L1 human

A list of points can approximate a separable space. Each point of the space has points of the list as close as you want. For $\mathbb{R}$, the list of all fractions does the job.