Defining a topological space on a set $X$ is equivalent to designating certain subobjects of $X$ in ${\bf Set}$ (monomorphisms into $X$ up to equivalence) as open. The requirements on the open sets of a topological space $X$ are equivalent to requiring the following:
$X \rightarrow X$ and $\emptyset \rightarrow X$ are open.
If $X_i \rightarrow X$ are open subobjects of $X$ for finite $i \in I$ then so is their product in the category of subobjects of $X$
If $X_i \rightarrow X$ are open subobjects of $X$ for $i \in I$, then so is their coproduct in the category of subobjects of $X$.
My question is about what happens when one dualizes this notion:
Let $X$ be a set and consider a subset of its quotient objects $\mathcal{S}$ such that:
$X \rightarrow X \in \mathcal{S}$ and $X \rightarrow \emptyset \in \mathcal{S}$.
If $X \rightarrow X_i \in \mathcal{S}$ for finite $i \in I$ then $\amalg_{i \in I} (X \rightarrow X_i) \in \mathcal{S}$, where the coproduct is taken in the category of quotient objects of $X$.
If $X \rightarrow X_i \in \mathcal{S}$ for $i \in I$, then $\prod (X \rightarrow X_i) \in \mathcal{S}$, where the product is taken in the category of quotient objects of $X$.
Does this structure arise anywhere in practice? What is known about this notion of 'co-topological spaces'? Is there a place I can learn about them?