"Weird-open" maps in topology

Question

Given topological spaces $X$ and $Y$, we define an open map from $X$ to $Y$ to be a map of sets $f\colon X\to Y$ satisfying the following condition:

• For each $U\in\mathcal{P}(X)$, if $U$ is open in $X$, then $f_*(U)$ is open in $Y$.

Here $f_*(U):=\{f(x)\in Y\ |\ x\in X\}$ is the direct image of $U$ by $f$, which sits in a triple adjunction $$f_*\dashv f^{-1}\dashv f_!\colon\mathcal{P}(X)\underset{\leftrightarrows}{\rightarrow}\mathcal{P}(Y),$$ where $f^{-1}$ is the inverse image, and where $f_!$, the “direct image with compact support”, is given by $$f_!(U):=\{y\in Y\ |\ f^{-1}(y)\subset U\}.$$ It can be sometimes useful to break down $f_!(U)$ into two sets: $$f_!(U):=f_{!,\mathrm{im}}(U)\cup f_{!,\mathrm{cp}}(U),$$ where \begin{align*} f_{!,\mathrm{im}}(U) &= f_!(U)\cap\mathrm{Im}(f),\\ f_{!,\mathrm{cp}}(U) &= f_!(U)\cap(Y\setminus\mathrm{Im}(f))\\ &= Y\setminus\mathrm{Im}(f). \end{align*} For example, if $f\colon\mathbb{N}\to\mathbb{N}$ is given by $f(n)=2n$, then $f_{!,\mathrm{im}}(U)=f_*(U)$ and $f_{!,\mathrm{cp}}(U)=\{\text{odd natural numbers}\}$ for $U\subset\mathbb{N}$.

Now, define a “weird-open map from $X$ to $Y$” as a map of sets $f\colon X\to Y$ satisfying the following condition:

• For each $U\in\mathcal{P}(X)$, if $U$ is open in $X$, then $f_!(U)$ is open in $Y$.

Since $f_!(U)$ contains $Y\setminus\mathrm{Im}(f)$, it's useful to also make the following relative definition: a “relatively weird-open map from $X$ to $Y$” as a map of sets $f\colon X\to Y$ such that:

• For each $U\in\mathcal{P}(X)$, if $U$ is open in $X$, then $f_!(U)\cap\mathrm{Im}(f)$ is open in $\mathrm{Im}(f)$.

Questions.

1. Is this an already studied notion (with a proper name)?

2. Are there any useful applications of it? (Be them in topology, analysis, algebraic geometry, etc.)

3. Regarding 2), an immediate property of $f_!$ that comes to mind is that it induces a further adjoint when passing to presheaves: a continuous map $f\colon X\to Y$ induces adjoint functors $$f^*\dashv f_*\dashv f_!\colon\mathsf{PSh}(Y)\underset{\leftrightarrows}{\rightarrow}\mathsf{PSh}(X),$$ and if $f$ is weird-open, there's an extra right adjoint $f_\dagger$ of $f_!$. (Here the relative notion is useful, for which we have a "locally defined adjoint" $f_\dagger\colon\mathsf{PSh}(X)\to\mathsf{PSh}(\mathrm{Im}(f))$.)

   Is $f_\dagger$ useful in practice?

Answer 1

As suggested in comments, I turn my comment into an answer here.

First of all let me note that in the overwhelming majority of texts I've seen notation is the opposite: $f_*$ from the OP is denoted by $f_!$ and $f_!$ by $f_*$. Still, to avoid further confusion I will stick to the notation of the question.

It is easy to check that for any subset $U$ of $X$ one has $f_!(U)=Y-f_*(X-U)$. It follows that $f$ is weird-open iff it is closed.

Accordingly, $f$ is relatively weird-open iff $f:X\twoheadrightarrow\operatorname{Im}(f)$ is closed.

As for presheaves, if $f$ is open but not necessarily continuous, the map $f_*:\operatorname{Opens}(X)\to\operatorname{Opens}(Y)$ has a right adjoint $f^*$ given by $f^*(V)=\operatorname{Interior}(f^{-1}(V))$ but $f^*$ might fail to have further right adjoint. Whereas if $f$ is continuous so that $f^*(V)=f^{-1}(V)$, but not necessarily open, then $f^{-1}:\operatorname{Opens}(Y)\to\operatorname{Opens}(X)$ may fail to have a left adjoint but it does have a right adjoint called direct image, sending $U$ to $\operatorname{Interior}(f_!(U))$. And if $f$ is closed then this is the same as $f_!(U)$.

Note also that for sheaf toposes (more generally, for geometric morphisms between Grothendieck toposes), preservation of some portion of colimits by the direct image is related to propriety (see the nLab entry for proper geometric morphisms). In particular, preservation of filtered colimits is called tidiness and preservation of joins of subterminals flatness. Existence of the right adjoint to the direct image is still stronger, such geometric morphisms are called local.