Can anyone help me understand this definition of manifold?

Question

This is the professor's definition of manifold after a one lecture introduction of topology. It is not precise, but I would like to understand it.

My question is:

1. in this definition, what is exactly the manifold? Is it the topological space M? Is it the mappings $phi_i$? Or is it sth else?

2. Which part of the definition implies that a manifold acts like a place locally?

Answer 1

1. A manifold is a pair $(M, \mathcal{A})$, where $M$ is a topological space (i.e.: a set with a topology) and $\mathcal{A}$ is an atlas. That is: $\mathcal{A}$ is a collection of charts (i.e.: homeomorphisms $\phi_i \colon V_i \to U_i$, where $V_i \subset M$ are open and $U_i \subset \mathbb{R}^n$ are open) which cover $M$ (i.e.: $M = \bigcup V_i$).

   The class that $M$ belongs to (the class of continuous manifolds, differentiable manifolds, smooth manifolds, etc.) depends on the overlap condition that the atlas $\mathcal{A}$ satisfies.

2. Manifolds $M$ have the property that they "locally look like $\mathbb{R}^n$" in the sense that the charts cover $M$. That is: Every point $p \in M$ on the manifold has a neighborhood $V \subset M$ which is homeomorphic (via a chart) to an open subset of $\mathbb{R}^n$.