a surface

A surface, in topology, is any shape that looks like a flat sheet of rubber if you zoom in close enough at any point — even though, zoomed out, it may curl into a sphere, loop into a doughnut, or twist into something stranger. The skin of a ball, the surface of a bagel, an infinite tabletop, the wall of a tube: every one is a surface, because a tiny ant standing anywhere on it would feel as if it were on an ordinary flat plane, with two independent directions to walk in. That 'locally two-dimensional and flat-looking' quality is the whole idea.

Made precise, a surface (a 2-manifold) is a topological space in which every point has a neighbourhood that is homeomorphic to an open disk of the plane — a small patch around each point can be deformed to a flat round piece of the plane. Surfaces come in two grand families. The compact ones without boundary — like the sphere and the torus — are finite and edgeless: walk forever and you never reach a rim. Surfaces with boundary have edges you can fall off, like a disk (its rim) or a cylinder (its two circular ends). The boundary, where it exists, is itself made of circles or lines.

Surfaces matter because they are the first interesting place topology can fully classify its objects: every compact surface without boundary is, up to homeomorphism, exactly one of a clean list — the sphere, the torus, the n-holed tori, and the non-orientable ones built from projective planes — sorted entirely by two simple numbers (how many holes, and whether it is orientable). The honest caution: a topological surface is not the same as a geometric surface with curvature. Topology sees a sphere and a wildly dented sphere as the very same surface, because it ignores curvature and shape and counts only holes and sidedness.