2manifolds

2-manifolds, commonly called surfaces, are topological spaces in which every point has a neighborhood homeomorphic to an open disk in the Euclidean plane. If the space is connected, it is a connected 2-manifold. A related notion is a 2-manifold with boundary, where boundary points have neighborhoods homeomorphic to a half-disk; the interior points satisfy the usual condition.

Compact connected 2-manifolds without boundary admit a complete classification: they are either orientable or non-orientable. Orientable closed surfaces are homeomorphic to a connected sum of g tori, denoted S_g, and include the sphere (g=0) and the ordinary torus (g=1). Non-orientable closed surfaces are homeomorphic to a connected sum of k projective planes, RP^2 # ... # RP^2, with k≥1; examples are the projective plane (k=1) and the Klein bottle (k=2).

Equivalently, such surfaces have Euler characteristic χ given by χ = 2 − 2g for orientable surfaces, and χ = 2 − k for non-orientable ones. This invariant together with orientability classifies compact closed 2-manifolds up to homeomorphism.

Beyond the closed case, compact 2-manifolds with boundary are classified by genus and boundary components, with invariants adjusted accordingly. Many 2-manifolds admit smooth structures and Riemannian metrics, and in complex analysis one obtains Riemann surfaces when a compatible complex structure is chosen on an orientable surface.