boundary in nLab

Contents

Contents

Idea

Loosely speaking, the boundary of a subset SS of topological space XX consists of those points in XX that are neither ‘fully in’ SS nor are ‘fully not in’ SS.

Definition

Of a subset of a topological space

For S⊂XS \subset X a subset of a topological space XX, the boundary or frontier ∂S\partial S of SS is its closure S¯\bar S minus its interior S ∘S^\circ:

∂S=S¯\S ∘ \partial S = \bar S \backslash S^\circ

Letting ¬\neg denote set-theoretic complementation, ∂S=¬(S ∘∪(¬S) ∘)\partial S = \neg (S^\circ \cup (\neg S)^\circ). It is a closed set. If we consider ∂\partial restricted to closed sets as an operation on closed sets, then it becomes a special case of the boundary operator on a co-Heyting algebra; see there for further properties.

Of a manifold

In a manifold with boundary of dimension nn the boundary is the collection of points which do not have a neighborhood diffeomorphic to an open n-ball, but do have a neighborhood homeomorphic to a half-ball, that is, an open ball intersected with closed half-space.

Properties

General

One reason behind the notation ∂\partial may be this (cf. co-Heyting boundary):

Notice the conclusion must fail if AA, BB are not closed, since in this case (∂A×B)∪(A×∂B)(\partial A \times B) \cup (A \times \partial B) is not closed (it doesn’t include ∂A×∂B\partial A \times \partial B).

Collar neighbourhood theorem

For topological manifolds and smooth manifolds with boudnary, see: collar neighbourhood theorem.

Some ramifications

The Leibniz rule shows that the boundary operator is better behaved when restricted to the lattice of closed subsets. Since this lattice forms a co-Heyting algebra, one is led to study algebraic operators axiomatizing properties of ∂\partial (Zarycki 1927) on these, the so called co-Heyting boundary operators.

Since the lattice of subtoposes of a given topos carries a co-Heyting algebra structure, it becomes possible to define (co-Heyting) boundaries of subtoposes and thereby even boundaries of the geometric theories that the subtoposes correspond to! Intuitively, such a boundary ∂T\partial T of a theory TT consists of those geometric sequents that neither ‘fully follow’ from TT nor ‘fully contradict’ TT.

The interior Int(ℰ j)Int(\mathcal{E}_j) of a subtopos ℰ j\mathcal{E}_j of a Grothendieck topos is defined in an exercise of SGA4 as the largest open subtopos contained in ℰ j\mathcal{E}_j. The boundary ∂ℰ j\partial\mathcal{E}_j is then defined as the subtopos complementary to the (open) join of the exterior subtopos Ext(ℰ j)Ext(\mathcal{E}_j) and Int(ℰ j)Int(\mathcal{E}_j) in the lattice of subtoposes.

The co-Heyting algebra perspective and the accompanying mereo-logic of theories was proposed by William Lawvere. See the references at co-Heyting boundary for further pointers!

Squaring things with algebraic topology

• interior

• co-Heyting boundary

• mereology

• boundary of a simplex

• corner

References

• M. Artin, A. Grothendieck, J. L. Verdier, Théorie des Topos et Cohomologie Etale des Schémas (SGA4), LNM 269 Springer Heidelberg 1972. (exposé IV, exercise 9.4.8, pp.461-462)

• M. Zarycki, Quelque notions fondamentales de l’Analysis Situs au point de vue de l’Algèbre de la Logique , Fund. Math. IX (1927) pp.3-15. (pdf)