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.


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.


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


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).


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).


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.


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


  • 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 August 19, 2015 04:46:34 by Thomas Holder (