Boundary (topology)

The boundary of a subset of a topological space is the set of points whose every neighborhood intersects both the subset and its complement.

Topological space: A set equipped with a collection of open subsets satisfying axioms for unions and finite intersections. Boundaries are defined relative to the open sets of an ambient space.
Derived set (topology): The set of accumulation points of a subset in a topological space. Boundary points are accumulation points of the set or of its complement.
Rational numbers: The real numbers expressible as ratios of integers with nonzero denominator. The rationals have the entire real line as their boundary in the usual topology.
Manifold boundary: The boundary points of a manifold modeled locally on a half-space rather than Euclidean space. This specialized boundary is not the same as the boundary of an arbitrary subset.
Neighborhood (topology): A set containing an open set around a point in a topological space. A point is on the boundary when every one of its neighborhoods meets both sides.
Frontier (topology): Another name for the boundary of a subset in a topological space. The frontier notation commonly expresses the boundary as closure minus interior.
Cantor set: The closed subset of the unit interval formed by repeatedly removing middle thirds. It is closed with empty interior, so every point in it lies on its boundary.
Topological boundary of a manifold: The boundary of a manifold regarded as a subset of an ambient topological space. It depends on the chosen ambient space, unlike the manifold's intrinsic boundary.
Closure (topology): The smallest closed set containing a given subset of a topological space. The boundary is the closure of a set with its interior removed.
Clopen set: A subset that is both open and closed in a topological space. A clopen subset has empty boundary, unlike sets with points separating inside from outside.

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