Questions about giving an example in subspace topology exercise

Question

I just want to know if my guess for the following question is correct"

Give an example of a separable Hausdorff space $(X,T)$ that has a subspace $(A, T_{A})$ is not separable.

I am guessing $X=\mathbb{R}$ and $A=\mathbb{N}$

Answer 1

A space $A$ is separable iff $A$ has a countable dense subset $D$. If $A$ is countable we may take $D=A.$ So any countable space is separable.

Properly speaking, a topological space is a pair $(X,T_X)$ where $T_X$ is a topology on $X.$ And if $A\subseteq X,$ then the subspace topology on $A$ (with respect to the topology $T_X$) is $\{t\cap A: t\in T_X\}.$

It is very common (& accepted) to speak of a space $X$ when what is meant is a space $(X,T_X),$ where the nature of $T_X$ may (or maybe not!) be understood from the context. Also common (& accepted) is to speak of a subspace $A$ of $X ,$ which is an abbreviation for (i) $A\subseteq X,$ and (ii) $T_X$ is some topology on $X,$ and (iii) the topology chosen for $A$ is $\{t\cap A: t\in T_X\}.$ The abbreviating is guaranteed to confuse at least a few students.

One example for your Q is the Niemitzky plane (a.k.a. the Moore plane or Moore-Niemitzky plane). See the definition in "Moore Plane" in Wikipedia. The countable set $\Bbb Q^2\cap (\Bbb R\times (0,\infty)\,)$ is dense. But the subspace $\Bbb R\times \{0\}$ is uncountable and discrete and hence not separable.

Answer 2

$\mathbb{N}$ in your idea is a countable subspace so automatically separable. For all metric spaces, separability is hereditary, being equivalent to second countability, and also in ordered spaces separability is hereditary (but not equivalent to second countability). So examples will be a bit "wild", though it can happen in TVS's, e.g.:

As a better example: by a standard theorem $X=\mathbb{R}^\mathbb{R}$ in the product topology is separable (e.g. the rational polynomials are countable and dense) and completely regular, but the subspace $A$ of all functions $\chi_x$ for $x \in \mathbb{R}$ (a function from $\Bbb R$ to itself that is $1$ for $x$, $0$ anywhere else for any fixed $x$) is uncountable and discrete so not separable.

A similar style example is the Sorgenfrey plane $\Bbb S^2$ (in Munkres $\Bbb R_l^2$) which has the antidiagonal $\{(x,-x): x \in \Bbb S\}$ as an uncountable discrete subspace and $\mathbb Q^2$ as a countable dense subset. Mrówka $\Psi$ and the Niemytzki plane, and the rational sequence topology are similar spaces. The phenomenon of separable spaces with large discrete subspaces is widespread.