A poset $(P,\le)$ can be turned into a category in a standard way. A group $(G,\cdot)$ can also be turned into a category in a standard way. Can we topological space $(X,\tau)$ into a category in a meaningful/useful way? In general, what pairs $(X,\eta)$ consisting of a set $X$ and a collection of subsets $\eta\subseteq\mathcal{P}(X)$ (with some properties) can be turned into a category?
I'm assuming that "meaningful way" means something like 1)you can recover the object from the category and 2)functors between these categories correspond to the appropiate morphisms between the objects
Thanks!