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I have an atlas $\{(U_1,\phi_1),(U_2,\phi_2)\}$ for $\mathbb{S}^1.$

$$U_1:=\{(Cos(2\pi t),Sin(2\pi t))|t \in (0,1)\},\\ \phi_1((Cos(2\pi t),Sin(2\pi t)):=t\in (0,1)\\ U_2:=\{(Cos(2\pi t),Sin(2\pi t))| t \in (-\frac{1}{2},\frac{1}{2})\}, \phi_2((Cos(2\pi t),Sin(2\pi t)):=t\in (-\frac{1}{2},\frac{1}{2}). $$

I am looking for an example of smooth (if possible) partition of unity $\{\alpha_{i}\}_{i \in \{1,2\}}$ over $\mathbb{S}^1$ such that $\alpha_1$ and $\alpha_2$ have compact support in $U_1$ and $U_2$, respectively.

Ted Shifrin
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chintan
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    Consider using this using 2 stereographic projections with convenient poles. – Jean Marie Apr 20 '24 at 09:37
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    @JeanMarie Small doubt: In that case, wouldn't the support for each function be entire circle and hence not contained in either U1 or U2? Thank you. – chintan Apr 20 '24 at 10:34

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Here's a suggestion: Think of this on the unit interval $[0,1]$ — ultimately you will identify $0$ and $1$. You have two open subsets, $V_1=(0,1)$ and $V_2=[0,1/2)\cup (1/2,1]$. Take a smooth function $f$ on $(0,1)$ with $f=1$ on $[3/8,5/8]$ and $f=0$ outside $(1/4,3/4)$. Now consider $f$ and $1-f$. $f$ will be your $\alpha_1$ and $1-f$ will be your $\alpha_2$. You can certainly write down a concrete formula, if you desire, by using the standard construction of bump functions.

Ted Shifrin
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