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I am familiar with the following notion of a shrinking or contraction lemma

Theorem: Given a complete metric space $(M,d)$ and a map $f$ into itself. If there exists a constant $K\in (0,1)$ such that $$d(f(x),f(y))\leq K d(x,y) $$ for any two $x,y\in M$, then $f$ has a unique fixed point $p\in M$. Moreover, the sequence of iterations of $f$ on any point $x_0\in M$ converges to $p$ (that is $\{(\underbrace{f\circ...\circ f}_{n\text{ times }})(x_0) \}_{n\in \mathbb{N}}\to p$).

Now, in A Comprehensive Introduction to Differential Geometry by M. Spivak, Theorem 13 states the following

Theorem (The Shrinking Lemma): Let $\mathcal{O}$ be an open locally finite cover of a manifold $\mathcal{M}$. Then it is possible to choose, for each $U$ in $\mathcal{O}$, an open set $U'$ with $U'\subset U$ in such a way that the collection of all $U'$ is also an open cover of $\mathcal{M}$.

Firstly, the proof is fairly easy to follow but an assumption is made, which I do not really get. Spivak claims that we can obviously assume that $\mathcal{M}$ is connected, which I think follows from that fact that any manifold is the disjoint union of countably many connected manifolds, but still is not that clear to me. Secondly (and I think this might be more interesting) what is exactly the relation of these two theorems which apparently hold the same name.

Thanks in advance

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Your version of the shrinking lemma may have a typo. As currently written, we can take $U' = U$. I think you mean we require $\overline{U'} \subset U$, where $\overline{U'}$ denotes the closure of $U'$. An open subset of a disjoint union is simply the disjoint union of open subsets of the base spaces. For example, $U\sqcup V \subset M\sqcup N$ is open if and only if $U, V$ are both open, this holds for a disjoint union over any indexing set. Thus once the theorem is proved in the connected case, the general case just follows by applying the theorem to each connected component, which is how the reduction was made.

The first theorem is the Banach fixed point theorem. It says that a contraction mapping in a complete metric space has a unique fixed point. It is used in various places in mathematics to prove the unique existence of something. For instance in Differential equations and Differential topology. The shrinking lemma on the other hand is used to prove that every open cover of a manifold has a partition of unity. It is much less ubiquitous within mathematics, I would say. The fact that they have similar names is a matter of chance (I would also say), in that they are fundamentally different.

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