$\newcommand{\R}{\mathbb{R}}$ I’ve always disliked the standard ways to define a manifold in differential geometry. First, the definition always starts with a topological space $M$. I don’t understand why you need make this assumption. I prefer to show that the topology of $M$ is a natural consequence of the definition. Second, the definition always uses two technical terms, paracompact and Hausdorff. I prefer to describe the properties concretely in terms of coordinate maps.

Below is how I prefer to define a manifold. It turns out that Peter Olver, in his book Applications of Lie Groups to Differential Equations (published in 1986) defines a manifold in exactly the same way (see Definition 1.1).

Start with a set $M$, just a set. Define a coordinate map to be a bijection $\phi: O \rightarrow \mathbb{R}^n$, where $O$ is a subset of $M$ and $\phi(O) \subset \mathbb{R}^n$ is open. Define an atlas to be a countable collection of coordinate maps, where the domains of the maps cover $M$. No assumptions about topology or smoothness yet.

A topological atlas is one where for any two coordinate maps $\phi_1: O_1 \rightarrow \mathbb{R}^n$ and $\phi_2: O_2\rightarrow \mathbb{R}^n$ such that $O_1\cap O_2 \ne \emptyset$, the change of coordinate map

\[\phi_2\circ\phi_1^{-1}: \phi_1(O_1\cap O_2) \rightarrow \phi_2(O_1\cap O_2)\]

is a homeomorphism. A topological manifold is a set $M$ with a topological atlas.

Observe that this defines a topology on $M$, where any coordinate map $\phi: O \rightarrow \phi(O) \subset \R^n$ is a homeomorphism. The assumption on the change of coordinate maps is exactly what is needed for this definition to be logically consistent.

Usually (but not always), there is one more assumption made, which I like to state as follows:

You can separate points using coordinate charts.

More precisely, given two different points $p_1, p_2 \in M$, there exist coordinate maps $\phi_1: O_1\rightarrow \R^n$, $\phi_2: O_2\rightarrow \R^n$ and open subsets $U_1 \subset O_1$, $U_2\subset O_2$ such that

  • $p_1 \in U_1$ and $p_2 \in U_2$
  • $U_1\cap U_2 = \emptyset$

The fact that the atlas has countably many coordinate maps is equivalent to $M$ being paracompact. The fact that points can be separated by coordinate charts is equivalent to $M$ being Hausdorff.

The definition of a smooth manifold is exactly the same, except that the change of coordinate maps are also assumed to be smooth, i.e., diffeomorphisms.