Definition (manifold):
Let
be a site, and let
be a subcategory of
. A manifold of type
consists of the aforementioned site
together with a class
of isomorphisms of
, where the
's are open in
and the
are objects of
, such that
- for all
there is a covering
of
such that for each
, there exists an
with
and
- whenever
st.
and
consitute a covering and
and
belong to the class
, the maps
and
guaranteed by the universal property of the pullback are isomorphisms onto their respective images and
is a morphism of
.
Proposition (fundamental induction lemma):
Let
be a logical statement, whose arguments are a topological manifold
and a closed subset
. Suppose that the following are true:
- Whenever
is compact and convex, where
is a coördinate chart on
, then
is true
- Whenever
and
are true, then
is true
- Whenever
is a descending chain of compact subsets of
,
is true for all
, then
is true
- Whenever
is true for all relatively compact, open
, then
is true
Then for all closed
,
is true.
Proof: First we prove by induction on
that for all sets of the type
for certain compact convex
(
), the statement
is true. We proceed by induction on
. For
, the statement is implied by the first assumption. Suppose now that
is true. Note that also
is true by the first assumption. Also
,
and
, where
is compact and convex as the intersection of two compact and convex sets. Thus, since the sets
are only
many, by induction on
, we may also conclude that
holds. By 2., we conclude
.
Now, we prove that
is true whenever
is a compact subset of a set of the type
, where
is compact and convex. Indeed, for each
, cover
by all the cubes of sidelength
centered at the points of
that intersect it. Then set
, so that
.
By the second assumption,
holds for each
, and hence by the third assumption
holds.
Now we claim by induction on
that whenever
are compact subsets of sets of the type
(
compact and convex), then
holds. For
, this follows from what we just proved. For the induction step, suppose that
holds. Note that also
holds by what we just proved. Then we have
,
and since
is a compact subset of
,
holds by what we just proved, and therefore, by induction,
holds. Hence, by 2., we get that
is true.
Now we are ready to prove that
is true whenever
is compact. Indeed, let
be compact. Then cover
by sets
, where
are some charts and
is compact and convex. By compactness of
, we may extract a finite subcover
. By intersecting with
and retaining compactness (as the intersection of two compact sets is compact), we may assume that
are contained within
and in particular
.
Thus, by the previous step,
holds.
Finally, let
be an arbitrary closed subset of
, and let
be a relatively compact open subset. Then
is true since
is compact as a closed subset of a compact set. Hence, by the fourth assumption,
is true.