Let's say $(X,\sigma)$ and $(X,\tau)$ are topological spaces, and $f$ is a continuous function from the former to the latter. (That is, the inverse images of elements of $\tau$ are elements of $\sigma$.) How would I write this? Saying:
Let $f:X\to X$ be continuous
clearly doesn't work, since they are meant to have separate topologies. However, saying:
Let $f:(X,\sigma)\to(X,\tau)$ be continuous
doesn't seem right, as the domain of $f$ is $X$, not the ordered pair $(X,\tau)$, and similarly for the codomain. Should I just do away with function notation completely, and say this?:
Let $f$ be a continuous function from the topological space $(X,\sigma)$ to the topological space $(X,\tau)$.
The same problem arises for metric spaces.