The following definition (theorem) is from "Calculus on Manifolds" by Michael Spivak.
Let $W$ be an arbitrary open set containing $A$.
Is $\Phi:=\{1:W\ni x\mapsto 1\in\mathbb{R}\}$ a $C^\infty$ partiion of unity for $A$?
If this is true, then we can very easily find a collection $\Phi$ satisfying (1) to (3) but it is not easy to find a collection $\Phi$ satisfying (1) to (4).
I feel strange.
Is the author's definition ok?
(1) For each $x\in A$ and for each $\varphi\in\Phi$, $0\leq\varphi(x)=1\leq 1$.
(2) For each $x\in A$, let $V:=W$. Then, $x\in V$ and the number of $\varphi\in\Phi$ which is not $0$ on $V$ is $1<\infty$.
(3) For each $x\in A$, we have $\sum_{\varphi\in\Phi}\varphi(x)=1(x)=1$.
3-11 Theorem.
Let $A\subset \Bbb R^n$ and let $\mathcal{O}$ be an open cover of $A$. Then there is a collection $\Phi$ of $C^\infty$ functions $\varphi$ defined in an open set containing $A$, with the following properties:(1). For each $x \in A$ we have $0 \leq \varphi(x) \leq 1$.
(2). For each $x \in A$ there is an open set $V$ containing $x$ such that all but finitely many $\varphi \in \Phi$ are $0$ on $V$.
(3). For each $x \in A$ we have $\sum_{\varphi \in \Phi}\varphi(x)=1$ (by (2) for each $x$ their sum is finite in some open set containing $x$).
(4). For each $\varphi \in \Phi$ there is an open set $U$ in $\mathcal{O}$ such that $\varphi = 0$ outside of some closed set contained in $U$.
(A collection $\Phi$ satisfying (1) to (3) is called a $C^\infty$ partiion of unity for $A$. If $\Phi$ also satisfies (4), it is said to be subordinate to the cover $\mathcal{O}$. In this chapter we will only use continuity of the functions $\varphi$.)
The theorem states that partitions of unity can be found for every open cover. You have just considered a very simple special case (in which, in fact, nothing has to be proven). In practice, one often applies the theorem to open covers with very many "small" open subsets. It won't be sufficient to just have the special case of a single big open subset.