Consider the category Top of topological spaces. Consider the contravariant functor from Top to Set sending a topological space X to the set of all opens of X. Is this functor representable?
What if you replace "open" by "closed"?
Consider the category Top of topological spaces. Consider the contravariant functor from Top to Set sending a topological space X to the set of all opens of X. Is this functor representable?
What if you replace "open" by "closed"?
Yes, it is (assuming the contravariant functor is what you mean, with the functoriality given by pull-back). Consider the topological space $T = \{0, 1\}$ with the open sets being $\{0\}$ and $T$. Then to give an open set in a topological space $X$ is the same as giving a map $X \to T$ (which sends said open set into ${0}$). The case for closed sets is the same.