see also algebraic topology, functional analysis and homotopy theory
Basic concepts
topological space (see also locale)
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
Examples
Basic statements
subsets are closed in a closed subspace precisely if they are closed in the ambient space
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Basic homotopy theory
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)