CW-complex, Hausdorff space, second-countable space, sober space
connected space, locally connected space, contractible space, locally contractible space
Loosely speaking, the boundary of a subset $S$ of topological space $X$ consists of those points in $X$ that are neither ‘fully in’ $S$ nor are ‘fully not in’ $S$.
For $S \subset X$ a subset of a topological space $X$, the boundary or frontier $\partial S$ of $S$ is its closure $\bar S$ minus its interior $S^\circ$:
Letting $\neg$ denote set-theoretic complementation, $\partial S = \neg (S^\circ \cup (\neg S)^\circ)$. 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.
In a manifold with boundary of dimension $n$ 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 $X, Y$ be topological spaces. Then for closed subsets $A \subseteq X$ and $B \subseteq Y$, the Leibniz rule $\partial (A \times B) = (\partial A \times B) \cup (A \times \partial B)$ holds.
Notice the conclusion must fail if $A$, $B$ are not closed, since in this case $(\partial A \times B) \cup (A \times \partial B)$ is not closed (it doesn’t include $\partial A \times \partial B$).
The interior operation preserves intersections, so $(A \times B)^\circ = ((A \times Y) \cap (X \times B))^\circ = (A^\circ \times Y) \cap (X \times B^\circ)$. Its complement is $(\neg A^\circ \times Y) \cup (X \times \neg B^\circ)$, whose intersection with $\widebar{A \times B} = A \times B$ is $(\partial A \times B) \cup (A \times \partial B)$.
If $A, B$ are connected open subsets of $X$ and $A \cap B$ is inhabited, then $\partial A = \partial B$ implies $A = B$.
Since $B$ is open, we have
where the right side is a disjoint union of open sets. $B$ is connected, so $B \subseteq A^\circ$ or $B \subseteq (\neg A)^\circ \subseteq \neg A$. The latter cannot occur since $A \cap B$ is inhabited. So $B \subseteq A^\circ \subseteq A$; by symmetry $A \subseteq B$.
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 $\partial T$ of a theory $T$ consists of those geometric sequents that neither ‘fully follow’ from $T$ nor ‘fully contradict’ $T$.
The interior $Int(\mathcal{E}_j)$ of a subtopos $\mathcal{E}_j$ of a Grothendieck topos is defined in an exercise of SGA4 as the largest open subtopos contained in $\mathcal{E}_j$. The boundary $\partial\mathcal{E}_j$ is then defined as the subtopos complementary to the (open) join of the exterior subtopos $Ext(\mathcal{E}_j)$ and $Int(\mathcal{E}_j)$ 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!
$H_n = Z_n/B_n$ | (chain-)homology | (cochain-)cohomology | $H^n = Z^n/B^n$ |
---|---|---|---|
$C_n$ | chain | cochain | $C^n$ |
$Z_n \subset C_n$ | cycle | cocycle | $Z^n \subset C^n$ |
$B_n \subset C_n$ | boundary | coboundary | $B^n \subset C^n$ |
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)