Manifold boundary ($\partial M$)

The manifold boundary of a topological manifold with boundary consists of the points which, in a half-space coordinate chart, have last coordinate zero. These are precisely the points without a neighbourhood homeomorphic to an open set in the full Euclidean space of the same dimension. It is an intrinsic property, independent of the coordinate chart. It differs from the boundary of a set in an ambient topological space: a circle has empty manifold boundary, but its boundary of a set in the Euclidean plane is the entire circle.