Here I begin to introduce the concept of a manifold, building on our intuition gained from studying topological spaces. I will formalise all of the terminology used in this video in future sections but for now I'll be speaking in fairly loose terms about manifolds.
We will see that the essential idea behind constructing a manifold is to take a topological space and cover it in charts. These charts are open sets which are homeomorphic to R^d and are viewed as giving coordinates to a local patch of the manifold. This can equally be viewed as saying that a manifold is a topological space which is locally homeomorphic to R^d.
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