TITLE: Open maps of spaces vs open maps of locales
QUESTION [6 upvotes]: A continuous function between topological spaces $f:X\to Y$ is called open, if $f[U]\in\mathcal{O}(Y)$ for all $U\in\mathcal{O}(X)$.
A morphism of locales $f:X\to Y$ is called open, if the associated morphism of frames $f^*:\mathcal{O}(Y)\to\mathcal{O}(X)$ has a monotone left adjoint $f_!$ which satisfies the Frobenius condition $f_!(U)\wedge V\leq f_!(U\wedge f^*(V))$ for all $U\in\mathcal{O}(X)$ and $V\in\mathcal{O}(Y)$.
It is easy to see that every continuous $f$ that is open in the topological sense is also open in the localic sense, but the converse is false in general. For example,  the maps $1\to\nabla(2)$ and $\mathsf{cofin}(\mathbb{N})\to\mathsf{sobrify}(\mathsf{cofin}(\mathbb{N}))$ are open in the localic, but not in the topological sense, since they're not surjective, but the induced locale-morphisms are iso.
These two examples involve non-sober spaces, so the natural thing to ask is: Do the two notions coincide for sober spaces? Or more generally, does the canonical functor from locales to spaces preserve open maps?
Or are there other criteria for when the two notions coincide? For example, I think I have convinced myself that continuous maps that are open in the localic sense are open in the topological sense whenever the codomain is $T_1$.

REPLY [4 votes]: Sketches of an Elephant notes (before Lemma C1.5.3)

Of course, we say a [locale morphism] $f: X \to Y$ is open if, for every open sublocale $U$ of $X$, its image $f_!(U)$ in $\mathrm{Sub}(Y)$ is an open sublocale of $Y$....this is not quite equivalent to the usual notion of openness for continuous maps of spaces; it is implied by the latter, but in the opposite direction it only implies that the set-theoretic image of each open subset of $X$ is 'almost open' in $Y$, in the sense that its subclosure is open. Even when X and Y are both sober, this does not suffice to make all such images open..., but it obviously does so if $Y$ is a $T_D$-space.

The distinction here seems to correlate with the distinction between subspaces and sublocales. The latter extends to a distinction between image of a subspace and image of a sublocale, which are invoked in the definition of open map of spaces and locales, respectively.