In the neighborhood formulation of topological spaces, the data of a topological space is a pair $(X,N)$ of a set $X$ and a function $N : X \to F X$, where $FX$ denotes the set of filters of subsets of $X$. We think of $N(x)$ as the filter of neighborhoods of the point $x \in X$. We can extend to an endofunctor $F: \mathbf{Set} \to \mathbf{Set}$ by setting, for any function $f: X \to Y$, a function $Ff: FX \to FY$, given by $$Ff(\phi) : = \{A \subseteq Y ~|~ f^{-1}( A) \in \phi\}~~~~.$$
A topological space is then a coalgebra for this endofunctor, subject to some further axioms.
Furthermore, in the neighborhood formulation, a function $f: X \to Y$ is continuous between topological spaces $(X, N_X)$ and $(Y, N_Y)$ if, for any $x \in X$ and $A \in N_Y(f(x))$, $f^{-1}(A) \in N_X(x)$. In our terms, this is simply $N_Y \circ f \leq Ff \circ N_X$. A continuous function $f: (X, N_X) \to (X, N_Y)$ is open if, for any $x \in X$ and $A \in N_X(x)$, there exists $B\in N_Y(f (x))$ such that $B \subseteq f (A)$. After some reflection, this turns out to be equivalent to $Ff \circ N_X \leq N_Y \circ f $. Thus a function is continuous and open if $N_Y \circ f = Ff \circ N_X$, i.e. if it is a coalgebra homomorphism.
It would be nice if this could be refined so as to recover the category of topological spaces and open maps as a category of coalgebras for a comonad. It looks like $F$ is not actually a comonad as defined, so we would need another version of $F$ to get off the ground. Does anyone have ideas, or am I barking up the wrong tree?