Given any map $f: X\to Y$ where $X$ is equipped with a topology $\tau$ we have a topology on $Y$ defined by $\{B\subset Y: f^{-1}(B)\in \tau\}$. This can be viewed as the inverse image of $\tau$ under the inverse image map of $f$: $f^{-1}: P(Y)\to P(X)$. I wonder if this topology has any significance on its own right. Note that this kind of construction can be done for any “weaker” structure than the lattice structure on $P(X)$. For example the following easy fact from the first chapter of Rudin’s Real and Complex Analysis:
Suppose $\mathfrak{M}$ is a $\sigma$-algebra in $X$, and $Y$ is a topological space. Let $f$ map $X$ into $Y$. If $\Omega$ is the collection of all sets $E\subset Y$ such that $f^{-1}(E)\in\mathfrak{M}$, then $\Omega$ is a $\sigma$-algebra in $Y$.
This can be used to show that for the $\sigma$-algebra $\mathfrak{N}$ generated by the topology on $Y$, any set $B\subset \mathfrak{N}$ has its inverse image measurable in $X$.
The inverse image function $f^{-1}: P(Y)\to P(X)$ is actually a lattice homomorphism (this means that arbitary intersection and union is distributive over the parenthesis(for example $\bigcup f^{-1}(A_i)=f^{-1}(\bigcup A_i)$), the completion can be moved outside the parenthesis, and $f^{-1}$ perserves subsets.)
So we have a functor from $\textbf{Set}$ to the category of (complete) lattices with the morphism mentioned in the above paragraph, defined by mapping a set $A$ to the power set lattice $P(A)$, and a set map $f: A\to B$ to the lattice homomorphism $f^*: P(B)\to P(A)$ defined by mapping a subset of $B$ to its inverse image under $f$.
Is there a name for this construction? Could anyone please provide a reference? Thank you.