Charts, transition maps, and smooth structure on differentiable manifolds
An n-dimensional manifold is a space that locally looks like ℝⁿ. A chart (U, φ) provides coordinates: U is an open neighborhood, φ maps it homeomorphically to an open subset of ℝⁿ. An atlas is a collection of charts covering the manifold. For a smooth manifold, all transition maps φβ∘φα⁻¹ must be smooth (C∞). The sphere S¹ needs ≥2 charts (stereographic projections). S² has a minimal atlas of 2 charts. The Möbius band is non-orientable — transition maps reverse orientation.