Boundary (topology)

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.

[1] Hausdorff also introduced the term residue, which is defined as the intersection of a set with the closure of the border of its complement.

Consequently, if these set are not empty[note 1] then they form a partition of

Conceptual Venn diagram showing the relationships among different points of a subset

area shaded green = set of interior points of

area shaded yellow = set of isolated points of

areas shaded black = empty sets.

The interior of the boundary of a closed set is empty.

[proof 1] 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.

the subset of rational numbers (whose topological interior in

Then These last two examples illustrate the fact that the boundary of a dense set with empty interior is its closure.

However, a closed subset's boundary always has an empty interior.

This example demonstrates that the topological boundary of an open ball of radius

(centered at the same point); it also shows that the closure of an open ball of radius

is not necessarily equal to the closed ball of radius

whose topology is equal to that induced by the (restriction of) the metric

Denote the open ball of radius

In particular, the open unit ball's topological boundary

And the open unit ball's topological closure

is a proper subset of the closed unit ball

that converges to it; the same reasoning generalizes to also explain why no point in

denotes the superset with equality holding if and only if the boundary of

The boundary operator thus satisfies a weakened kind of idempotence.

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.