B From local to global

In this chapter, we gather some topological preliminaries allowing to build global objects from local ones. This is usually not discussed in informal expositions where such arguments are either implicit or interspersed with more specific arguments.

We first need to discuss how to build a function having everywhere some local properties from a sequence of functions having those properties on bigger and bigger parts of the source space. We actually want to also accommodate finite sequences so we start with a definition of the source of our sequences.

Definition B.1

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For every natural number $N$ we set

\[ \mathcal{I}_{N} = \begin{cases} ℕ \text{ if $N = 0$}\\ \{ 0, \dots , N - 1\} \text{otherwise} \end{cases} \]

On each $\mathcal{I}_{N}$ we use the obvious linear ordering. In particular there is no maximal element when $N = 0$ and $N-1$ is maximal if $N$ is positive. The successor function $S \! :\mathcal{I}_{N} → \mathcal{I}_{N}$ is the function sending $n$ to $n+1$ unless $n$ is maximal, in which case $S(n) = n$.

Our first lemma gives a criterion ensuring that a sequence of functions is locally ultimately constant hence has a limit that locally ultimately agrees with the elements of the sequence. Remember that a family of sets $V_n$ in a topological space $X$ is locally finite if every point of $X$ has a neighborhood that intersects only finitely many $V_n$.

Lemma B.2

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Let $X$ be a topological space and let $Y$ be any set. Let $f$ be a sequence of functions from $X$ to $Y$ indexed by $\mathcal{I}_{N}$ for some $N$. Let $V$ be a family of subsets of $X$ indexed by $\mathcal{I}_{N}$ such that, for every non-maximal $n$, $f_{S(n)}$ coincides with $f_n$ outside $V_{S(n)}$. If $V$ is locally finite then there exists $F \! :X → Y$ such that, for every $x$ and every sufficiently large $n$, $F$ coincides with $f_n$ near $x$.

Proof ▶

The assumption that $V$ is locally finite gives, for every $x$ in $X$, a subset $U_x$ of $X$ such that $U_x$ is a neighborhood of $x$ and intersects only finite many $V_n$’s. In particular we can find an upper bound $n₀(x)$ of the set of indices $n$ in $\mathcal{I}_{N}$ such that $V_n$ intersects $U_X$. Since, for every non-maximal $n$, $f_{S(n)}$ coincides with $f_n$ outside $V_{S(n)}$, we get by induction that, for all $n ≥ n₀(x)$, $f_n$ coincides with $f_{n₀(x)}$ on $U_x$.

We now define $F$ as $x ↦ f_{n₀(x)}(x)$. We claim that, for every $x$, $F$ coincides with $f_n$ on $U_x$ as soon as $n$ is at least $n₀(x)$. Indeed let us fix $x$ and $n ≥ n₀(x)$ and $y ∈ U_x$. We have

\begin{align*} f_n(y) & = f_{n₀(x)}(y) \text{ since $n ≥ n₀(x)$ and $y ∈ U_x$}\\ & = f_{\max (n₀(x), n₀(y))}(y) \text{ since $\max (n₀(x), n₀(y)) ≥ n₀(x)$ and $y ∈ U_x$}\\ & = f_{n₀(y)}(y) \text{ since $\max (n₀(x), n₀(y)) ≥ n₀(y)$ and $y ∈ U_y$}\\ & = F(y) \text{ by definition of $F$}. \end{align*}

In the preceding lemma, the limit function $F$ inherits all local properties of the elements of the sequence. In order to make this precise, we need the language of germs of functions. One can define germs with respect to any filter but we will need only the case of neighborhood filters : two functions $f$ and $g$ define the same germ at some point $x$ if they coincide near $x$.

Definition B.3

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Let $X$ be a topological space, $x$ a point in $X$ and $Y$ a set. A germ of function from $X$ to $Y$ at $x$ is an element of the quotient $(X → Y)_x$ of the set of functions from $X$ to $Y$ by the relation $f ∼ g$ if $f$ and $g$ coincide near $x$. The image of a function $f$ in this quotient will be denoted by $[f]_x$.

A local predicate on functions from $X$ to $Y$ is a family $P$ of predicates on the germ set $(X → Y)_x$ for every $x$ in $X$. We say that a function $f$ satisfies $P$ at $x$ if $P [f]_x$ holds, and $f$ satisfies $P$ if it satisfies $P$ at every point.

For instance if $Y$ is also equipped with a topology then continuity is (equivalent to) a local predicate on functions from $X$ to $Y$ since a function is continuous if and only if it is continuous at every point $x$ and this condition only depends on the germ of the function at $x$.

We also need to build local predicates by localizing some local predicates near some subsets.

Definition B.4

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Let $X$ be a topological space, $A$ a subset of $X$, $Y$ a set and $P$ a local predicate on functions from $X$ to $Y$. The restriction of $P$ to $A$ is the local predicate $P_{|A}$ defined by the constraint that a function $f$ satisfies $P_{|A}$ at $x$ if $x ∈ A$ implies that $f$ satisfies $P$ near $x$.

Note the above definition hides a little lemma asserting that the obtained predicate is indeed local. An even smaller lemma asserts that a function satisfies $P_{|A}$ if and only if it satisfies $P$ at each point near $A$.

In the next lemma, there are three predicates or families of predicates. The local predicate $P₀$ is satisfied by every function appearing in the lemma, it could be a continuity or smoothness constraint. The family of local predicates $P₁$ is the main constraint and the goal is to build a function satisfying all of them. The family of predicates $P₂$ plays an auxilliary role, it does not have to be local, does not appear in the conclusion and is only used to bring more flexibility in the main inductive assumption. One can read “$f$ satisfies $P₂^i$” as “$f$ can be improved in $Uᵢ$”.

Lemma B.5

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Let $X$ be a topological space and $Y$ be any set. Let $U$ be a locally finite family of subsets of $X$ indexed by some $\mathcal{I}_{N}$. Let $P₀$ be a local predicate on functions from $X$ to $Y$, let $i ↦ P₁^i$ be a family of such predicates, and let $i ↦ P₂^i$ be a family of predicates on functions from $X$ to $Y$, all families being indexed by $\mathcal{I}_{N}$. Assume that

there exists $f₀ \! :X → Y$ satisfying $P₀$ and $P₂^0$ ;
for every $i$ in $\mathcal{I}_{N}$ and every $f \! :X → Y$ satisfying $P₀$, $P₂^i$ and every $P₁^j$ for $j {\lt} i$, there exists a function $f' \! :X → Y$ which coincides with $f$ outside $U_i$ and satisfies $P₀$ and every $P₁^j$ for $j ≤ i$ as well as $P₂^{S(i)}$ unless $i$ is maximal.

Then there exists $f \! :X → Y$ which satisfies $P₀$ and all $P₁^i$’s.

Proof ▶

The main assumption from the lemma allows to build by induction a sequence $f$ of functions from $X$ to $Y$ indexed by $\mathcal{I}_{N}$ such that, for every $n ∈ \mathcal{I}_{N}$,

$f_n$ satisfies $P₀$
for every $i ≤ n$, $f_n$ satisfies $P₁^i$.
$f_{S(n)}$ satisfies $P₂$ unless $n$ is maximal.
$f_{S(n)}$ coincides with $f_n$ outside $U_{S(n)}$.

Note that the first term of this sequence isn’t $f₀$ but the function obtained by applying the induction assumption to $f₀$.

The preceding lemma applied to this sequence gives a map $f$ which locally coincides with every element which is far enough in the sequence. Let $x$ be a point in $X$. Let $n$ be large enough to ensure $f$ coincides with $f_n$ near $x$. By definition this means $[f]_x = [f_n]_x$ and we know $P₀ [f_n]_x$ hence we get $P₀ [f]_x$. Now fix also $n$ in $\mathcal{I}_{N}$. Let $n'$ be large enough to be larger than $n$ and such that $[f]_x = [f_{n'}]_x$. Since $n' ≥ n$ we have $P₁^n [f_{n'}]_x$ hence $P₁^n [f]_x$.

Next we will need a version of the above lemma building a homotopy of maps. In this version, $P₀$ is still a predicate such as continuity satisfied by all maps from $X$ to $Y$ entering the discussion. Then $P₀'$ is analogous but for maps from $ℝ × X$ to $Y$, and it will come with some affine invariance assumption ensuring its compatibility with concatenation of homotopies. Instead of having a completely general family of local predicates $P₁^i$, we fix a single local predicate $P₁$ but it will be required to hold only near some subset $K_i$ (as in Definition B.4).

Homotopies of maps from $X$ to $Y$ are usually meant to be continuous maps from $[0, 1] × X$ to $Y$. In a differential topology context, one requires smoothness and in order to be able to easily concatenate homotopies, it is very convenient to add the assumptions that those maps are independent of the time variable $t ∈ [0, 1]$ when $t$ is close to $0$ or $1$. Especially in a formalization context, it is even more convenient to assume homotopies are defined on $ℝ × X$, and time independent near $(-∞, 0] × X$ and $[1, +∞) × X$. Continuity or smoothness don’t appear in the following abstract lemma where they are replaced by arbitrary local predicates.

Lemma B.6

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Let $X$ be a topological space and $Y$ be any set. Let $P₀$ and $P₁$ be local predicates on maps from $X$ to $Y$. Let $P₀'$ be a local predicate on maps from $ℝ × X → Y$. Assume that for every $a$, $b$ and $t$ in $ℝ$, every $x$ in $X$ and every $f \! :ℝ × X → Y$, if $f$ satisfies $P₂$ at $(at + b, x)$ then $(t, x) ↦ f(at+b, x)$ satisfies $P₀'$ at $(t, x)$. Let $f₀ \! :X → Y$ be a function satisfying $P₀$ and such that $(t, x) ↦ f₀(x)$ satisfies $P₀'$.

Let $K$ and $U$ be families of subsets of $X$ indexed by some $\mathcal{I}_{N}$. Assume that $U$ is locally finite and $K$ covers $X$.

Assume that, for every $i$ in $\mathcal{I}_{N}$ and every $f \! :X → Y$ satisfying $P₀$ and satisfying $P₁$ on $\bigcup _{j {\lt} i} K_j$, there exists $F \! :ℝ × X → Y$ such that

for all $t$, $F(t, \cdot )$ satisfies $P₀$
$F$ satisfies $P₀'$
$F(1, \cdot )$ satisfies $P₁$ on $\bigcup _{j ≤ i} K_j$
$F(t, x) = f(x)$ whenever $x$ is not in $U_i$ or $t$ is near $(-∞, 0]$
$F(t, x) = F(1, x)$ whenever $t$ is near $[1, +∞)$.

Then there exists $F \! :ℝ × X → Y$ such that

for all $t$, $F(t, \cdot )$ satisfies $P₀$
$F$ satisfies $P₀'$
$F(0, \cdot ) = f₀$
$F(1, \cdot )$ satisfies $P₁$.

Proof ▶

Carefully checking all details is a bit technical but the strategy is as follows. We fix an increasing sequence $T \! :\mathcal{I}_{N} → [0, 1)$ starting at $0$, say $i ↦ 1 - 1/2^i$. We want to build a sequence of homotopies $F_i \! :ℝ × X → Y$ where each $F_i$ is time-independent on $[T_i, + ∞) × X$ and, assuming $i$ isn’t maximal, $F_{S(i)}$ is built from $F_i$ by applying the induction assumption to $F_i(T_i, \cdot )$ and rescaling the obtained homotopy by the affine map sending $[0, 1]$ to $[T_i, T_{S(i)}]$.

Hence we want to apply Lemma B.5 with source space $\hat{X} = ℝ × X$. We use as the background local predicate $\hat{P}₀$ at $(t, x)$ the constraint on a function $F$ that $F(t, \cdot )$ satisfies $P₀$ at $x$ and $P₀'$ at $(t, x)$ and if $t = 0$ then $F(t, x) = f₀(x)$. As the target family of local predicates $\hat{P}₁$ we use for every $i ∈ \mathcal{I}_{N}$ the constraint on $F$ at $(t, x)$ that if $t = 1$ and $x$ is near $K_i$ then $F(t, \cdot )$ should satisfy $P₂$ at $x$. As the auxiliary family of predicates $\hat{P₂}$ at index $i$ we use the constraint of being time-independent on $[T_i, + ∞) × X$.

In order to explain how the induction assumption of the current lemma implies the induction assumption of Lemma B.5, we fix $i ∈ \mathcal{I}_{N}$ and a map $F \! :ℝ × X → Y$ that satisfies $\hat{P}₀$, is time-independent on $[T_i, + ∞) × X$ and satisfies $\hat{P}₁^j$ for all $j {\lt} i$. By this last requirement and the time independence property, we get that $F(T_i, \cdot )$ satisfies $P₁$ near $\bigcup _{j {\lt} i} K_j$. Our induction assumption applied to $F(T_i, \cdot )$ then gives $F' \! :ℝ × X → Y$ such that

for all $t$, $F'(t, \cdot )$ satisfies $P₀$
$F'$ satisfies $P₀'$
$F'(1, \cdot )$ satisfies $P₁$ on $\bigcup _{j ≤ i} K_j$
$F'(t, x) = f(x)$ whenever $x$ is not in $U_i$ or $t$ is near $(-∞, 0]$
$F'(t, x) = F(1, x)$ whenever $t$ is near $[1, +∞)$.

As the new map required by the inductive assumption of Lemma B.5, we pick

\[ F'' \! :(t, x) ↦ \begin{cases} F(t, x) \text{ if $t ≤ T_i$}\\ F’\left((t - T_i)/(T_{S(i)} - T_i), x\right) \text{ if $t {\gt} T_i$} \end{cases} \]

Fully checking that $F''$ is suitable is fairly technical but mostly straightforward. Care is required in particular to check that $F''$ coincides with $F$ near $(T_i, x)$ for every $x$. This uses both the fact that $F$ is time-independent on $[T_i, + ∞) × X$ and that $F'$ is time-independent near $(-∞, 0] × X$ hence in particular near $(0, x)$.

In a different direction, we need a version of Lemma B.5 where we do not fix any family of subsets of the source space, but simply want to derive existence of a function satisfying some local predicates from the assumptions of existence of local solution and the ability to patch solutions. This requires putting a lot more constraints on the source topological space in order to use the following classical result.

Lemma B.7

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Let $X$ be a metrizable locally compact second countable topological space. Let $C$ be a closed subset in $X$. Let $P$ be a non-decreasing predicate on subsets of $X$ (meaning that if $U ⊂ V$ and $V$ satisfies $P$ then so does $U$). Assume the empty set satisfies $P$ and every point in $C$ has a neighborhood in $X$ satisfying $P$. Then there exist sequences of subsets $K$ and $W$ indexed by natural numbers such that $K$ covers $C$, $W$ is locally finite and, for every $n$ :

$K_n$ is compact
$W_n$ is open
$K_n ⊂ W_n$
$W_n$ satisfies $P$.

Proof ▶

This is a classical result.

In the next lemma, $P₀$ is again a background local predicate satisfied by all maps entering the discussion, and $P₁$ is the main target local predicate. We also use an extra predicate $P₀'$ that enters the patching assumption in an asymmetric way and will allows to deduce a relative version of the lemma.

Lemma B.8

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Let $X$ a second countable locally compact metrizable topological space. Let $P₀$, $P₀'$ and $P₁$ be local predicates on function from $X$ to a set $Y$. Let $f₀ \! :X → Y$ be a function satisfying $P₀$ and $P₀'$. Assume that

For every $x$ in $X$, there exists a function $f \! :X → Y$ which satisfies $P₀$ and satisfies $P₁$ near $x$.
For every closed subsets $K₁$ and $K₂$ of $X$ and every open subsets $U₁$ and $U₂$ containing $K₁$ and $K₂$, for every function $f₁$ and $f₂$ satisfying $P₀$, if $f₁$ satisfies $P₀'$ and satisfies $P₁$ on $U₁$ and if $f₂$ satisfies $P₁$ on $U₂$ then there exists $f$ which satisfies $P₀$ and $P₀'$, and satisfies $P₁$ near $K₁ ∪ K₂$ and coincides with $f₁$ near $K₁ ∪ U₂^c$.

Then there exists $f$ which satisfies $P₀$, $P₀'$ and $P₁$.

Proof ▶

The assumptions on the topology of $X$ and local existence of solutions allow to apply Lemma B.7 to get sequences of subsets $K$ and $U$ of $X$ indexed by natural numbers such that $K$ covers $X$, $U$ is locally finite and, for every $i$:

$K_i$ is compact
$U_i$ is open
$K_i ⊂ U_i$
there is a function $f \! :X → Y$ which satisfies $P₀$ and satisfies $P₁$ on $U_i$.

We then apply Lemma B.5 to the family of subsets $U$ with local predicates $\hat{P}₀$ combining $P₀$ and $P₀'$ and $\hat{P}₁^i$ asking that $P₁$ holds near $K_i$, and the trivial family of auxiliary predicates $\hat{P}'₀$.

For this we need to explain how the patching assumption of the current lemma implies the induction assumption of Lemma B.5. So we fix an index $i ∈ ℕ$ and a function $f$ which satisfies $\hat{P}₀$ and satisfies all $\hat{P}₁^j$ for $j {\lt} i$. We denote by $K$ the closed subset $\bigcup _{j {\lt} i} K_j$ and denote by $V$ an open neighborhood of $K$ such that $f$ satisfies $P₁$ on $V$. The patching assumption applied to $K$, $K_i$, $V$ and $U_i$ with functions $f$ and the local solution on $U_i$ gives a suitable new function.

From the above lemma we can deduce a version with only a base local predicate $P₀$ and a target one $P₁$ and starting from a function which is already good near some closed subset $K$. The is the version that is actually used in our application.

Lemma B.9

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Let $X$ a second countable locally compact metrizable topological space. Let $P₀$ and $P₁$ be local predicates on functions from $X$ to a set $Y$. Let $K$ be a closed subset of $X$. Let $f₀ \! :X → Y$ be a function satisfying $P₀$ and satisfying $P₁$ near $K$. Assume that

For every $x$ in $X$, there exists a function $f \! :X → Y$ which satisfies $P₀$ and satisfies $P₁$ near $x$.
For every closed subsets $K₁$ and $K₂$ of $X$ and every open subsets $U₁$ and $U₂$ containing $K₁$ and $K₂$, for every function $f₁$ and $f₂$ satisfying $P₀$, if $f₁$ satisfies $P₁$ on $U₁$ and if $f₂$ satisfies $P₁$ on $U₁$ then there exists $f$ which satisfies $P₀$, and satisfies $P₁$ near $K₁ ∪ K₂$ and coincides with $f₁$ near $K₁ ∪ U₂^c$.

Then there exists $f$ which satisfies $P₀$ and $P₁$ and coincides with $f₀$ near $K$.

Proof ▶

We reduce this to Lemma B.8 using as auxilliary local predicate $P₀'$ the constraint to coincide with $f₀$ near $K$.

Our patching condition almost matches the one from Lemma B.8 except that each $(K₁, K₂, U₁, U₂)$ should be replaced by $(K ∪ K₁, U ∪ U₁, K₂, U₂)$ where $U$ is a suitable neighborhood of $K$.