In Definition 2.1 on page 68 and 69 Glen E. Bredon defines a $n$-differentiable manifold as a second countable Hausdorff space $M^n$ and a collection of charts such that:
a chart is a homeomorphism $\phi:U\to U'\subset\mathbb{R}^n$ where $U$ is open in $M^n$ and $U'$ is open in $\mathbb{R}^n$;
each point $x\in M$ is in the domain of some chart;
- for charts $\phi:U\to U'\subset \mathbb{R}^n$ and $\psi:V\to V'\subset\mathbb{R}^n$ we have that the “change of coordinates” $\phi\psi^{-1}:\psi(U\cap V)\to\phi(U\cap V)$ is $C^\infty$; and
- the collection of charts is maximal with properties 1, 2, and 3.
I'm confused about property 2 where he talks about a point $x\in M$ being element of an open subset of $M^n$.
In definition 2.4 on the next two pages he talks about an functionally structured Hausdorff space $(F,M^n)$ where he also talks about points $x\in M$ being elements of open subsets of $M^n$.
My guess is that he writes $M^n$ to imply that he talks about an $n$-differentiable manifold or is he talking about an embedding of $M$ in $M^n$?
From the comment above by @Max.
You are right, the notation $M^n$ is used to indicate that $M$ is an $n$-dimensional differentiable manifold. It is not used in the sense of the Cartesian product $M \times \dots \times M$.