Say we have two topological spaces $(X, \tau_X )$, $(Y, \tau_Y )$. As I understand it, a homeomorphism between two topological spaces is an isomorphism $\phi : X\rightarrow Y$, but both $\phi$ and $\phi^{-1}$ have the additional property of being continuous, which means that the pre-image of open sets in the codomain are open i.e. $\forall V\in \tau_Y $ we have $\phi^{-1}(V) \in \tau_X$ and the image of open sets in the domain is open i.e. $\forall U \in \tau_X$ we have $\phi(U)\in \tau_Y$.
As far as I can tell, it is the continuity of $\phi$ which makes it respect the additional structure of topologies. But does the continuity imply that the topologies $\tau_X$ and $\tau_Y$ are isomorphic as sets? Quoting wikipedia:
Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space.
I mean surely two isomorphic sets with isomorphic topologies are topologically equivalent, right? I'm just wondering if the converse to this statement is true?
Thanks for any replies!