Any function $f\colon A\to B$ defines a triple adjunction $f_*\dashv f^{-1}\dashv f_!$ between the powersets $\mathcal{P}(A)$ and $\mathcal{P}(B)$, where \begin{align*} f_*(U) &:= \{b\in B\ |\ \text{there exists $a\in U$ such that $f(a)=b$}\}\\ f^{-1}(V) &:= \{a\in A\ |\ f(a)\in V\}\\ f_!(U) &:= \{b\in B\ |\ f^{-1}(b)\subset U\} \end{align*} with $U\in\mathcal{P}(A)$ and $V\in\mathcal{P}(B)$. Now, we may define open, closed, and continuous maps using these:
- A map $f$ is open if $f_*$ sends opens to opens.
- A map $f$ is continuous if $f^{-1}$ sends opens to opens.
- A map $f$ is closed if $f_!$ sends opens to opens (see the proof by მამუკა here).
We could now repeat this procedure with relations, although this time the triple adjunction breaks down into two adjunctions: any relation $R\colon A ⇸ B$ defines two adjunctions $R_*\dashv R_{-1}$ and $R^{-1}\dashv R_!$, where \begin{align*} R_*(U) &:= \{b\in B\ |\ \text{there exists $a\in U$ such that $b\in R(a)$}\}\\ R_{-1}(V) &:= \{a\in A\ |\ R(a)\subset V\}\\ R^{-1}(V) &:= \{a\in A\ |\ R(a)\cap V\neq\emptyset\}\\ R_!(U) &:= \{b\in B\ |\ R^{-1}(b)\subset U\} \end{align*} with $U\in\mathcal{P}(A)$ and $V\in\mathcal{P}(B)$.
Note: A nice fact here is that $R_{-1}=R^{-1}$ iff $R$ is total and functional, i.e. $R^{-1}$ and $R_{-1}$ coincide precisely if $R$ comes from a function.
Mimicking the situation for functions, we could now make the following definitions:
- A relation $R$ is open if $R_*$ sends opens to opens.
- A relation $R$ is strongly continuous if $R_{-1}$ sends opens to opens.
- A relation $R$ is weakly continuous if $R^{-1}$ sends opens to opens.
- A relation $R$ is closed if $R_!$ sends opens to opens (A very similar argument to the one given by მამუკა for functions shows that this is the same as asking $R_*$ to send closed sets to closed sets).
Weakly continuous relations have apparently been studied before (see this PlanetMath page), and one nice property of them is that an equivalence relation $\mathord{\sim}$ on a topological space $X$ is weakly continuous iff the projection map $\pi\colon X\twoheadrightarrow X/\mathord{\sim}$ is an open map.
There is, however, an issue: continuous maps can be equivalently defined as those $f$ for which $f$ sends closed sets to closed sets, which follows from the equality $A\setminus f^{-1}(V)=f^{-1}(B\setminus V)$. Now, this equality doesn't need to hold for either $R_{-1}$ or $R^{-1}$, as we have \begin{align*} R_{-1}(B\setminus V) &= \{a\in A\ |\ R(a)\subset B\setminus V\},\\ A\setminus R_{-1}(V) &= \{a\in A\ |\ R(a)\not\subset V\},\\ R^{-1}(B\setminus V) &= \{a\in A\ |\ R(a)\setminus V\neq\emptyset\},\\ A\setminus R^{-1}(V) &= \{a\in A\ |\ R(a)\cap V=\emptyset\}. \end{align*} Considering also relations $R$ for which $R^{-1}$ or $R_{-1}$ send closed sets to closed sets thus leads to a total of four different definitions of continuity for relations using this approach.