nLab
boundary

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

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 SXS \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

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

Proposition

Let X,YX, Y be topological spaces. Then for closed subsets AXA \subseteq X and BYB \subseteq Y, the Leibniz rule (A×B)=(A×B)(A×B)\partial (A \times B) = (\partial A \times B) \cup (A \times \partial B) holds.

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).

Proof

The interior operation preserves intersections, so (A×B) =((A×Y)(X×B)) =(A ×Y)(X×B )(A \times B)^\circ = ((A \times Y) \cap (X \times B))^\circ = (A^\circ \times Y) \cap (X \times B^\circ). Its complement is (¬A ×Y)(X׬B )(\neg A^\circ \times Y) \cup (X \times \neg B^\circ), whose intersection with A×B¯=A×B\widebar{A \times B} = A \times B is (A×B)(A×B)(\partial A \times B) \cup (A \times \partial B).

Proposition

If A,BA, B are connected open subsets of XX and ABA \cap B is inhabited, then A=B\partial A = \partial B implies A=BA = B.

Proof

Since BB is open, we have

B¬B=¬A=A (¬A) ,B \subseteq \neg \partial B = \neg \partial A = A^\circ \cup (\neg A)^\circ,

where the right side is a disjoint union of open sets. BB is connected, so BA B \subseteq A^\circ or B(¬A) ¬AB \subseteq (\neg A)^\circ \subseteq \neg A. The latter cannot occur since ABA \cap B is inhabited. So BA AB \subseteq A^\circ \subseteq A; by symmetry ABA \subseteq B.

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

H n=Z n/B nH_n = Z_n/B_n(chain-)homology(cochain-)cohomologyH n=Z n/B nH^n = Z^n/B^n
C nC_nchaincochainC nC^n
Z nC nZ_n \subset C_ncyclecocycleZ nC nZ^n \subset C^n
B nC nB_n \subset C_nboundarycoboundaryB nC nB^n \subset C^n

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)

Revised on April 29, 2017 04:27:27 by Urs Schreiber (92.218.150.85)