In a topological space, the set of points each of whose every neighbourhood intersects both the set and its complement; equivalently the closure of the set minus its interior.

Principle

Boundary points are precisely those that are neither wholly interior nor wholly exterior; they mark the interface between a set and its complement and are invariant under taking closure and complement operations (∂A = cl(A) \, int(A)).

Demonstration

For the interval [0,1] in the real line with standard topology, the boundary is {0,1}; for the open unit disk in the plane, the boundary is the unit circle, capturing the geometric edge.

Misapplication

Confusing boundary with closure or frontier in a metric sense (assuming boundary points must be accumulation points) can mislead: isolated points of a finite set are boundary points in a discrete topology but not accumulation points.

Consequence

Correct identification of the boundary supports arguments about continuity, compactness, integration domains, and solving PDE boundary-value problems; it determines where values or conditions must be prescribed for well-posedness.

Reversal

Taking the complement focuses on exterior points rather than the interface: studying just the complement's interior misses the structure of points that lie exactly at the interface between set and complement.

Boundary

Defined in any topological space; the concept excludes measure-theoretic boundaries (which may coincide but are distinct concepts) and properties that require metrizability such as distances to the boundary. The boundary can be empty (for whole space or empty set) or equal to the set (for closed nowhere-dense sets).

Semantic Tension

Tension exists with the measure-theoretic or topological frontier notions and with the concept of 'edge' in geometric contexts; 'boundary' is topological and invariant under homeomorphism, while geometric edge notions may depend on embedding or metric.

Synthesis

The boundary of a set is the topological interface: points that neither belong to a purely interior neighbourhood of the set nor of its complement, forming the locus where topological and often analytic conditions must be specified.