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 $x$ this 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 author wrote "by (2) for each $x$ their sum is finite in some open set containing $x$".
What does "by (2) for each $x$ this sum is finite in some open set containing $x$" mean?
(a) Does this mean $\sum_{\varphi\in\Phi} \varphi(y)<\infty$ for any $y$ in some open set containing $x$ for each $x$?
But by (3), for each $x \in A$ we have $\sum_{\varphi \in \Phi}\varphi(x)=1<\infty$.
So, I don't think this means $\sum_{\varphi\in\Phi} \varphi(y)<\infty$ for any $y$ in some open set containing $x$ for each $x$.
(b) Does this mean $\#\{\varphi\in\Phi:\varphi(y)\neq 0\}<\infty$ for any $y$ in some open set containing $x$ for each $x$?
But by (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 $V$.
So, for any $x\in A$, $\#\{\varphi\in\Phi:\varphi(x)\neq 0\}<\infty$.
So, I don't think this means $\#\{\varphi\in\Phi:\varphi(y)\neq 0\}<\infty$ for any $y$ in some open set containing $x$ for each $x$.
Please explain what "by (2) for each $x$ this sum is finite in some open set containing $x$" means.
Spivak actually means (a), $\sum_{\varphi\in\Phi} \varphi(y)<\infty$ for any $y$ in some open set $V_x$ containing $x$, since by (2) there is an open set $V_x$ containing $x$ such that all but finitely many $\varphi \in \Phi$ vanish (this is (b)), and those which don't vanish are bounded. Actually, (3) is a corollary of (2) because we can normalize each $\varphi \in \Phi$ by redefining $\varphi'(x) =\varphi(x)/\sum_{\varphi\in\Phi} \varphi(x)$ for each $\varphi \in \Phi$.