What is the subobject classifier in the category of topological spaces, if it exists?
I think it's the indiscrete space on two points, since continuous maps from a space $X$ to this space correspond to arbitrary subsets of $X$. However, I saw a claim that it was in fact the Sierpinski space.
The indiscrete space on two points only classifies the regular subobjects, which correspond to the subspaces here. The Sierpinski space on two points classifies the open subspaces, and the discrete space on two points classifies the clopen subspaces.
In a category with a subobject classifier, every monomorphism $U \to X$ is regular (it is the equalizer of $\chi_U,\chi_X : X \rightrightarrows \Omega$). In particular, the category has to be balanced (since any regular monomorphism which is an epimorphism must be an isomorphism). The existence of non-homeomorphic continuous bijections shows that $\mathbf{Top}$ is not balanced.
Background on topos theory. The lack of a subobject classifier means that $\mathbf{Top}$ is not a topos. However, the indiscrete space on two points classifies strong subobjects. While the other axioms of a topos are not satisfied either, they are satisfied for other variants: pseudotopological spaces, diffeological spaces, quasi-topological spaces each form a quasitopos.
