Work In Progress!
A manifold in the broadest sense, is a structure which is locally homeomorphic to $\mathbb{R}^n$ at each of its open sets. In layman’s terms, this means that while a manifold may be sufficiently “abstracted” such that its elements look nothing like those of $\mathbb{R}^n$, we can regardless treat the manifold as if it’s $\mathbb{R}^n$, with a few potential caveats.