What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful?


I have a vague idea of the possibility of using continuous relations in science, for example in biochemistry or celestial mechanics, as a less deterministic approach.
I have never seen anything about this topic on internet. I have an opinion of my own, but would like to hear what professional mathematicians think about it.



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> Can it be adequately defined? And if so, would it possibly be of any
>  use at all?
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Conditions on the definition:


1. Composition of continuous relations should be continuous.
2. Partial functions, continuous on its domain, should be continuous.
3. Given a partial function $f:X\times Y \rightarrow Z$ that is continuous on its domain, 
then $R=\{(x,y)\in X\times Y|f(x,y)=z\_0\}$ should be a continuous relation.



Continuous relations?
Here's a different and quite generic approach: Let $X,Y$ be topological spaces. Then we topologize $\mathcal{P}(Y)$ and say that $R\subseteq X\times Y$ is continuous if and only if the function $f\_R: X \to \mathcal{P}(Y)$ defined by $x\mapsto \{y\in Y: (x,y) \in R\}$ is continuous.


As for topologizing $\mathcal{P}(Y)$ you can take the topology generated by $\{\mathcal{S} \subseteq \mathcal{P}(Y): (\bigcup \mathcal{S}) \textrm{ is open in }Y\}$. Possibly other topologies on $\mathcal{P}(Y)$ are more natural.


