I am reading "Calculus on Manifolds" by Michael Spivak.
Spivak wrote "Since $\{D_1, \dots, D_n\}$ covers $A$, we have $\psi_1(x) + \dots + \psi_n(x) > 0$ for all $x$ in some open set $U$ containing $A$".
Spivak didn't write "Since $\{\text{int } D_1, \dots, \text{int } D_n\}$ covers $A$, we have $\psi_1(x) + \dots + \psi_n(x) > 0$ for all $x$ in $\text{int } D_1 \cup \dots \cup \text{int } D_n$ which is open and contains $A$".
Why?
And I think it is possible to take $\mathbb{R}^n$ as the domain of $\phi$'s.
But Spivak didn't take $\mathbb{R}^n$ as the domain of $\phi$'s.
Why?
Spivak was certainly right, but why?
