Answer to Question #217685 in Differential Geometry | Topology for Prathibha Rose

Question

Give an example of seperable Hausdroff space which has a non seperable subspace

Answer 1

Consider $\mathbb{R}$ with the lower-limit topology, i.e. the topology generated by the base $\{[a,b)|a,b\in \mathbb{R} \text{ and } a<b\}$. If we generalize this topology on $\mathbb{R}^2$, the topology we'll get is finer than the usual topology on $\mathbb{R}^2$. So it is Hausdorff and separable.

$\{(x,-x)|x \in \mathbb{R}\}$ is an uncountable discrete subspace of this topological space.