Defining continuity for functions between surfaces

Question

In general, considering function $f: M_{1} \rightarrow M_{2}$ between 2 manifolds, how does one formalize the idea of that function being continuous? Specifically, I am asking this in the context of needing to prove that if a neighbourhood of a point $x$ in the first manifold is in its' interior, then $f(x)$ is in the interior of $M_{2}$

Answer 1

Well both $M_1$ and $M_2$ are both topological spaces, so continuity of a function $f : M_1 \to M_2$ means the usual topological definition of continuity, i.e. $f$ is continuous if for every open set $U$ of $M_2$ we have $f^{-1}[U]$ to be an open subset of $M_1$.

Furthermore if $x$ is an interior point of $M_1$, then $x$ is contained in some chart $(U, \phi)$ where $U$ is an open set of $M_1$ and $\phi : U \to \phi[U] \subseteq \mathbb{R}^{2}$ is a homeomorphism and where $\phi[U]$ is an open subset of $\mathbb{R}^2$.

To show that $f(x)$ is an interior point of $M_2$ you need to show that $f(x)$ is contained in some chart $(V, \psi)$ where $V$ is an open set of $M_2$ and $\psi : V \to \psi[V] \subseteq \mathbb{R}^{2}$ is a homeomorphism for which $\psi[V]$ is an open subset of $\mathbb{R}^2$.