differentiable manifold

A differentiable manifold locally looks like Euclidian space. A topological space \((X, \mathcal{T})\) is locally Euclidian if for \(x\in X\), there is an open neighborhood containing \(x\), \(U \in \mathcal{T}\) and there is a homeomorphism \(h\) that takes \(U\rightarrow U'\), where \(U'\) is an open neighborhood of \(\mathbb{R}^n\). Then, for functions on \(U\) we can do calculus by going to \(U'\) and then coming back.