I have troubles in a little part of the following proposition.

Let $(X,\tau)$ be a topological space and $\Im=\left\{U_{\alpha}\right\}_{\alpha \in I}$ an open cover of $X$. If $\Im$ has a locally finite cozero-set refinement, then there exists a locally finite partition of unity subordinate to $\Im$.

Here is my attempt

Let $\vartheta:=\left\{V_{\beta}\right\}_{\beta \in J}$ be a locally finite cozero-set refinement of $\Im$.

By hipothesis $\forall \beta \in J$ , $V_{\beta}$ is a cozero-set in $X$ $\Rightarrow$ for each $\beta \in J$ there exists an $f_{\beta}\in C(X)$ such that $V_{\beta}=X-Z(f_{\beta})$ (Where $Z(f_{\beta})$ is the zero-set of $f_{\beta}$).

Define $f:(X,\tau)\rightarrow (\mathbb{R},\tau_{u})$ by the formula

$f(x)=\sum_{\beta \in J}f_{\beta}(x)$ for each $x\in X$

By hipothesis $\vartheta$ is locally finite in $X$ and $f_{\beta}$ is a continuous function for each $\beta \in J$ $\Rightarrow$ $f\in C(X)$.

Now for each $\beta \in J$, define $\phi_{\beta}=f_{\beta}-f$.

Then I want to show that $\left\{\phi_{\beta}\right\}$ is a locally finite partition of unity subordinate $\Im$.

$\left\{\phi_{\beta}\right\}$ es locally finite in $X$ because $P:=\left\{P_{\beta}\right\}_{\beta \in J}$ where $P_{\beta}=X-Z(f_{\beta})=V_{\beta}$ is locally finite in $X$ and $P$ subordinate to $\Im$ because $P<\Im$

Here is my problem I can´t show that $\left\{\phi_{\beta}\right\}$ is a partition of unity of $X$. I can´t show that

$\sum_{\beta \in J}\phi_{\beta}(x)=1$

¡I need a hint please!


In the standard construction we additionally impose that each function $\{f_\beta\}$ maps the space $X$ into the segment $[0,1]$ and then we define $\phi_b=f_\beta/f$.


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