In topology and mathematics in general, the boundary of a subset of a topological space is the set of points in the closure of not belonging to the interior of . An element of the boundary of is called a boundary point of . The term boundary operation refers to finding or taking the boundary of a set. Notations used for boundary of a set .

Terminology

Some authors (for example, Willard in General Topology) use the term frontier instead of boundary in an attempt to avoid confusion with a different definition used in algebraic topology and the theory of manifolds. Despite widespread acceptance of the meaning of the terms boundary and frontier, they have sometimes been used to refer to other sets. For example, Metric Spaces by E. T. Copson uses the term boundary to refer to Hausdorff's border, which is defined as the intersection of a set with its boundary. Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.

Definitions

There are several equivalent definitions for the boundary is understood: It is the closure minus the interior denotes the closure denotes the topological interior with the closure of its complement such that every neighborhood which are not in either the interior or exterior

A boundary point defined above is sometimes called the set's topological boundary to distinguish it from other similarly named notions such as the boundary of a manifold with boundary or the boundary of a manifold with corners, to name just a few examples.

A connected component of the boundary of is called a boundary component of .

Examples

Consider the real line with the usual topology (that is, the topology whose basis sets are open intervals the subset of rational numbers (whose topological interior we have

These last two examples illustrate the fact that the boundary of a dense set to be non-empty. However, a subset's boundary always has an empty interior.

, its boundary is the closed interval is given its usual topology, the subspace topology itself, its boundary is empty.

Properties

The boundary of a set is closed;

A set is closed if and only if it contains its boundary, and open if and only if it is disjoint from its boundary.

("Trichotomy") and these three sets are pairwise disjoint.

contains at least one point in the set and at least one point not in the set. The boundary of the interior of a set as well as the boundary of the closure of a set are both contained in the boundary of the set.

is a dense open

The interior of the boundary of a closed set is empty. Consequently, the interior of the boundary of the closure of a set is empty. The interior of the boundary of an open set is also empty. This fact is important for the definition and use of nowhere dense subsets, meager subsets, and Baire spaces.

A set is the boundary of some open set if and only if it is closed and nowhere dense. The boundary of a set is empty if and only if the set is both closed and open (that is, a clopen set).

Conceptual Venn diagram = set of accumulation points set of boundary points area shaded green = set of interior points area shaded yellow = set of isolated points is either an accumulation point or an isolated point. Isolated points are always boundary points.

Boundary of a boundary

where \,\supseteq\, denotes the superset The boundary operator thus satisfies a weakened kind of idempotence.

In discussing boundaries of manifolds or simplexes and their simplicial complexes, one often meets the assertion that the boundary of the boundary is always empty. Indeed, the construction of the singular homology rests critically on this fact. The explanation for the apparent incongruity is that the topological boundary (the subject of this article) is a slightly different concept from the boundary of a manifold or of a simplicial complex. For example, the boundary of an open disk viewed as a manifold is empty, as is its topological boundary viewed as a subset of itself, while its topological boundary viewed as a subset of the real plane is the circle surrounding the disk. Conversely, the boundary of a closed disk viewed as a manifold is the bounding circle, as is its topological boundary viewed as a subset of the real plane, while its topological boundary viewed as a subset of itself is empty. In particular, the topological boundary depends on the ambient space, while the boundary of a manifold is invariant.

See also

• See the discussion of boundary in topological manifold for more details.

• Exterior (topology) Largest open set disjoint from some given set

• , for measure-theoretic characterization and properties of boundary

Notes

Citations

References