Why don't we ever speak of the continuity of charts on a manifold?

Question

A topological manifold $M$ requires that all of its transition functions are continuous. Why don't we instead or additionally require that the charts themselves be continuous. After-all, the manifold is a topological space, and the base space (usually $\mathbb{R}$) is also a topological space, so we can speak of continuity for any chart $(\mathcal{U},\phi)\in\mathscr{A}$: $$ \phi\in C^0(\mathcal{U},\mathbb{R}^n)\quad :\Leftrightarrow\quad \forall V\in\mathbb\tau_{{R}^n}: \mathrm{preim}_\phi(V)\in\tau_{M} $$ Why don't we ever speak of this? Is it possible to have continuous transition functions but non-continuious charts?

Answer 1

We do!

For example, often times you see the definition that a locally Euclidean space is a topological space, which is locally homeomorphic to $\mathbb{R}^n$. Then you might further read that a topological manifold is a locally Euclidean space, that is Hausdorff and second countable. Sometimes second countability is omitted. Charts are never explicitly mentioned.