TITLE: Does Every Subset of a Separable Topological Space have Countably Many Isolated Points?
QUESTION [5 upvotes]: This is almost certainly a duplicate, but I keep seeing this result on metric spaces, not topological ones.
Let $(X,\tau)$ be a topology. A set $A\subset X$ is dense if $A\cap B\neq\emptyset$ for all $B\in\tau$. We say $(X,\tau)$ is separable if there exists a countable, dense $A\subseteq X$.
Given some $A\subseteq X$, a point $p\in A$ is an isolated point in $A$ if there exists $O\in\tau$ such that $p\in O$ and $O\cap A=\{p\}$.
I am wondering: If $X$ is separable and $A\subseteq X$, then must the set of isolated points in $A$ be at most countable? Perhaps if we add the condition that it is Hausdorff it is true.
My attempt: If $A$ has $0$ or $1$ isolated points, we are done. Otherwise, let $p_{1},p_{2}\in A$ be isolated points of $A$. Then there exist $O_{1},O_{2}\in\tau$ such that $O_{1}\cap A=\{p_{1}\}$ and $O_{2}\cap A=\{p_{2}\}$. Furthermore, because $(X,\tau)$ is Hausdorff, there exist $T_{1},T_{2}\in\tau$ such that $p_{1}\in T_{1},p_{2}\in T_{2}$, and $T_{1}\cap T_{2}=\emptyset$.
Now, because open sets are closed under finite intersection, we have that $O_{1}\cap T_{1}$ and $O_{2}\cap T_{2}$ are open, disjoint sets which have intersection $\{p_{1}\}$ and $\{p_{2}\}$ with $A$, respectively.
My idea from here is to well order some countable dense subset and use the well-ordering to choose one element from each open set around each isolated point (without using choice because we can just choose the least element). But I have yet to show that there exists a collection of disjoint open sets, one for each isolated point. I am not sure how to continue.
For example, the result is true in the reals for closed sets by Cantor-Bendixon (I think). However the proof I saw was nothing like this and the fact that I've not seen a more general statement for any set of reals seems like an indicator that it is not true. Is it true if I add more restrictions? Maybe a stronger separation axiom?

REPLY [8 votes]: A simple example is the Sorgenfrey plane, i.e., the plane $\mathbb R\times\mathbb R$ with the topology generated by the half-open rectangles $[a,b)\times[c,d)$. The set $\mathbb Q\times\mathbb Q$ of all rational points is a countable dense set, and the anti-diagonal $\{(x,-x):x\in\mathbb R\}$ is an uncountable discrete closed subset.
Another nice example is the compact Hausdorff space $\{0,1\}^\mathfrak c$, the product of continuum many two-point discrete spaces, which can  be shown to be separable. The set of all points with a single nonzero coordinate is an uncountable discrete subset.