Vtyjkyuk

I need help in a passage of a proof.

I want to show :

Let $M$ be a smooth manifold, and let $U_alpha_alpha$ be an open cover of $M$. Then exists a partition of unity subordinate to $U_alpha$, i.e. a collection $rho_alphacolon Mto mathbbR _alpha$ of $C^infty$ functions such that:

1) for each $alpha$ we have Image$rho_alphasubseteq[0,1]$

2) for each $alpha$ we have supp$rho_alphasubseteq U_alpha$

3) for each point $pin M$ exists an open neighborhood $A_p$ of $p$ such that $A_p cap$ supp$rho_alphaneemptyset$ at most for finitely many indices $alpha$.

4) for each point $pin M$ we have $sum_alphamid rho_alpha(p)ne0rho_alpha(p)=1$.

Proof.

Let $M$ be a smooth manifold, and let $U_alpha_alpha$ be an open cover of $M$. I already know that exists $Vj$ and $Wj$ countable open cover of $M$, such that

1) for each $j$, $overlineV_j$ is compact and $overlineV_jsubseteq W_j$

2) $W_j$ is a locally finite refinement of $U_alpha$, i.e. each $W_j$ is contained in some $U_alpha$ and for each point $pin M$ exists an open neighborhood $A_p$ of $p$ such that $A_p cap W_jneemptyset$ at most for finitely many indices $j$.

For each $j$ let $phi_jcolon M to mathbbR$ be a smooth bump function for the couple $(overlineV_j,W_j)$, i.e.

1) $phi_j$ is $C^infty$ in $M$

2) Im$phi_j subseteq [0,1]$

3) supp$phi_j subseteq W_j$

4) $phi_j|_overlineV_jequiv 1$.

Let $phi colon M to mathbbR$ be such that for each $p in M$ we have $phi (p)=sum_j: pin Wj phi_j(p)$.

Then, $phi$ is $C^infty$ and $phi(p)ge 1$ for every $p in M$.

Finally, let $rho_alpha colon M to mathbbR$ be such that

$$rho_alpha(p)=fracsum_j : W_j subseteq U_alpha textand ; pin W_j phi_j(p)phi(p)$$

I want to show that $rho_alphacolon Mto mathbbR _alpha$ verify the properties 1) 2) 3) 4) of the yellow rectangle.

I have problem with showing property 3)

If I am correct, i have shown that $pin M mid rho_alpha(p)ne 0=bigcup_j: W_j subseteq U_alpha pin M mid phi_j(p)ne 0$ and taking the closure in $M$ i have supp$rho_alpha=bigcup_j: W_j subseteq U_alpha$ supp$phi_j subseteq bigcup_j: W_j subseteq U_alpha W_j subseteq U_alpha$.

P.S. I don't know if $j: W_j subseteq U_alpha$ is finite or non, but i can take the closure and preserve the "$=$" in the equation
$$pin M mid rho_alpha(p)ne 0=bigcup_j: W_j subseteq U_alpha pin M mid phi_j(p)ne 0$$
even if in general "closure of union $ne$ union of closures" if the union is infnite. This because of the following Lemma: if $mathfrakX$ is a locally finite collection of subsets of a topolofical space $M$, then $$overlinebigcup_XinmathfrakXX=bigcup_XinmathfrakXoverlineX$$

But I'm being unable to show property 3).

Thank you for your help.

Edit

Here is my reasoning.
Suppose that $p$ is a point in $M$ such that $rho_alpha(p)ne 0$ for some index $alpha$. This means that there is an index $j$ such that $W_j subseteq U_alpha$ and $phi_j(p)ne 0$. Let $A:= betamid W_j subseteq U_beta$. Then $Ane emptyset$ since $alpha in A$. We have $rho_beta(p)ne0 ; forall beta in A$. So, if $A$ is infinite, then $pin$ supp$rho_beta ;forall beta in A$ and so it is not possible that property 3 is true. So I think I should prove that $A$ must be finite, but $U_alpha$ is an arbitrary open cover of $M$, so why should $A$ be finite?

I'm in total confusion. Maybe the proof is wrong?

Edit n.2 , 17/05/2018

I'm having doubts also for property 4).

Suppose that $rho_alpha(p)ne 0$ for some index $alpha$. This means that there is an index $j$ such that $W_jsubseteq U_alpha$ and $phi_j(p)ne0$. If $betanealpha$ is another index such that also $W_jsubseteq U_beta$, then the addend $phi_j(p)$ appears also in $rho_beta (p)$ and so $phi_j(p)$ appears at least twice in $$sum_gammamid rho_gamma(p)ne0rho_gamma(p)$$ and this would imply that $$sum_alphamid rho_alpha(p)ne0rho_alpha(p)ge1$$ So also property 4) is not true.

Where does my reasoning falls?

Thank-you for your help.