The antidiagonal $\{(x,-x)\}$ is an uncountable discrete subspace of the Sorgenfrey plane, so having a countable dense subset is not a hereditary property

Refuted: that the property "has a countable dense subset" is hereditary (Hereditary, open-hereditary and closed-hereditary properties of topological spaces, Dense, nowhere dense and codense subsets of a topological space, and the criterion by basic open sets).

Witness. In the Sorgenfrey plane $S\times S$ (The Sorgenfrey plane: the product of two half-open-interval lines has the rectangles $[a,b)\times [c,d)$ as a basis and $\mathbb{Q}\times \mathbb{Q}$ as a countable dense subset), which has the countable dense subset $\mathbb{Q}\times \mathbb{Q}$, take the antidiagonal

$L:=\{(x,-x):x\in \mathbb{R}\}$

with the subspace topology (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace). Then:

1. $L$ is discrete: for every $x$, the basic rectangle $[x,x+1)\times [-x,-x+1)$ meets $L$ exactly in $\{(x,-x)\}$, so every singleton of $L$ is open in $L$ and the subspace topology is the discrete one (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies).
2. $L$ is uncountable (Finite, countably infinite, countable, uncountable), being in bijection with $\mathbb{R}$ ($\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874)).
3. The only dense subset of $L$ is $L$ itself, since in a discrete space every subset is closed. So $L$ has no countable dense subset, although the space it sits inside has one.

The word separable is not used: it is not defined at this point in the reading order, and the three claims above say in full what it would abbreviate.

• The rectangle in step 1.1 is chosen with both corners at the point. Any basic rectangle $[x,b)\times [-x,d)$ with $b>x$ and $d>-x$ would do, and the computation is the same: the first factor forces $t\ge x$ and the second forces $t\le x$. It is the half-openness on the left in both coordinates, together with the reversal of the sign in the second, that isolates the point.

• The property is open-hereditary, and that is the sharp statement. By claim 4 of For $A\subseteq S\subseteq X$ the closure of $A$ in $S$ is ${\bar{A}}^X\cap S$, while the interior only contains ${\mathrm{int}}^X(A)\cap S$, with equality when $S$ is open; and a dense subset of $X$ traces to a dense subset of every open $S$ a dense subset of a space traces to a dense subset of every open subspace, so "has a countable dense subset" passes to open subspaces. The antidiagonal is not open in $S\times S$, and the failure above shows that the hypothesis in that claim cannot be dropped.

• Nothing here needs a choice principle. The surjection $L\to \mathbb{R}$ is written down, the rectangles are written down, and the only nonconstructive ingredient is $\mathbb{R}$ is uncountable (Cantor's nested intervals, 1874), whose own proof is choice free.