I am reading "Analysis on Manifolds" by James R. Munkres.
Theorem 16.3 (Existence of a partition of unity). Let $\mathcal{A}$ be a collection of open sets in $\mathbb{R}^n$; let $A$ be their union. There exists a sequence $\phi_1,\phi_2,\dots$ of continuous functions $\phi_i:\mathbb{R}^n\to\mathbb{R}$ such that:
(1) $\phi_i(x)\geq 0$ for all $x$.
(2) The set $S_i=\operatorname{Support}\phi_i$ is contained in $A$.
(3) Each point of $A$ has a neighborhood that intersects only finitely many of the sets $S_i$.
(4) $\sum_{i=1}^{\infty}\phi_i(x)=1$ for each $x\in A$.
(5) The functions $\phi_i$ are of class $C^{\infty}$.
(6) The sets $S_i$ are compact.
(7) For each $i$, the set $S_i$ is contained in an element of $\mathcal{A}$.
A collection of functions $\{\phi_i\}$ satisfying conditions (1)-(4) is called a partition of unity on $A$. If it satisfies (5), it is said to be of class $C^\infty$; if it satisfies (6), it is said to have compact supports; if it satisfies (7), it said to be dominated by the collection $\mathcal{A}$.
DOUBT : I wonder why the author wrote $\phi_i:\mathbb{R}^n\to\mathbb{R}$ instead of $\phi_i:A\to\mathbb{R}$.
Of course if the functions $\phi_i:\mathbb{R}^n\to\mathbb{R}$ satisfy (1)-(7), then the functions $\phi_i|A:A\to\mathbb{R}$ satisfy (1)-(7).
And,
Of course if the functions $\psi_i:A\to\mathbb{R}$ satisfy (1)-(7), then the functions $\phi_i:\mathbb{R}^n\to\mathbb{R}$ such that $\phi_i(x):=\psi_i(x)$ if $x\in A$ and $\phi_i(x):=0$ if $x\in\mathbb{R}^n-A$ satisfy (1)-(7).
When we want to know why the Author wrote something is a particular way , we have to take the (w)holistic view : What were the Axioms the Author used , What were the Definitions used , How were earlier theorems developed , Where will the current theorem get used ....
Here there are a couple of reasons why it is $\mathbb{R}^n$ , not $A$ :
(I) Criteria (1) says "for all $x$" , where "for all $x \in \mathbb{R}^n$" is implicit. When the functions are defined/available only in $A$ , then what we can say about the other values outside of $A$ ? It will be undefined/unavailable.
(II) Criteria (2) says the Support is contained in $A$ , whereas the Previous Definition about Support says "If $\phi:\mathbb{R}^n \rightarrow \mathbb{R}$ , then Support of $\phi$ is defined to be the Closure of the Set $\{x: \phi(x) \not = 0\}$" : naturally , when we use this Definition , we must have $\phi$ over $\mathbb{R}^n$ , not only $A$.
(III) Criteria (4) says that the Sum is $1$ over $A$. The functions have values over the rest of $\mathbb{R}^n$ where the Sum might or might not add up to $1$.
(IV) Definition of Class $C^{\infty}$ was earlier using $\mathbb{R}^n$ , though the Author later shows that it works even with SubSpaces.
(V.1) Proof uses $\psi_i:\mathbb{R}^n \rightarrow \mathbb{R}(x)$ , hence theorem must match that.
These $\psi_i$ are non negative over all $x \in \mathbb{R}^n$
Using these , the Author gets $\lambda(x)$ which will converge in $A$ , though we do not know [ & do not bother ! ] whether it will converge over $\mathbb{R}^n$
(V.2) Proof uses Lemma 16.1 & Lemma 16.2 , which require $\mathbb{R}^n$ , hence this theorem uses $\mathbb{R}^n$ to use the earlier concepts & works.
(VI) It will be tautological & redundant to say that $\mathcal{A}$ will "Dominate" the functions over $\mathcal{A}$ ....
It is more interesting & meaningful to claim that $\mathcal{A}$ will "Dominate" the functions over $\mathbb{R}^n$ with the given Criteria ....
(VII) We have to then move on with the text to see where this theorem is used & whether it requires $\mathbb{R}^n$ [ it will ! ] & whether we can tweak it like OP did.
SUMMARY :
Author uses $\mathbb{R}^n$ because it is necessary due to the way the textbook was written & due to the way the concepts were elaborated.