Understanding Partition of Unity

I am following the proof of Riesz Representation Thm, where $$\textbf{"subordinate partition of unity"}$$ is used. However, I have problem in understanding the definition.

Could there be any (possibly better geometric) elaboration on the definition (Especially if we assumed below $$X=\mathcal{C}[X]$$ ) .

This is the def. I read;

$$\textbf{Definition}$$: Let $$\{U_{\alpha}\}_{\alpha \in I}$$ be an open cover of a metric space X. A partition of unity subordinates to $$\{U_{\alpha}\}_{\alpha \in I}$$, is $$\{\Psi_{U_{\alpha}}\}_{\alpha \in I}$$ , such that $$\Psi_{U_{\alpha}} \in \mathcal{C}(X) ,\ 0\leq \Psi_{U_{\alpha}} \leq \mathcal{X}_{[U_{\alpha}]} \ \text{and} \ \sum_{\alpha\in I} \Psi_{U_{\alpha}}(x) = 1$$

All this means is that for each $$\alpha\in I,$$ the support of $$\Psi_{U_{\alpha}}$$ is contained in $$U_{\alpha},\$$ the values of each $$\Psi_{U_{\alpha}}$$ lie in the interval $$[0,1]$$ and for each $$x\in X$$, the sum $$\sum_{\alpha\in I} \Psi_{U_{\alpha}}$$ is actually finite and equal to $$1$$.
The geometric idea is that you have a bunch of functions, each one of which is concentrated in a covering element of the metric space $$X$$. This allows you, in many cases, to move from the local to the global: if you have for example, smooth functions defined only in a neighborhood of a given point, by using the partition of unity, you can cobble them together to get a globally defined function that retains the local property.
• so is this right? $\to$ let's say that X is compact. we can take any bunch of functions on X, and write a collection of functions $\Psi_{U_\alpha} : U_\alpha\to [0,1]$ such that $\forall x\in X , \sum_\alpha \Psi_{U_\alpha}(x) = 1$. Sep 29, 2019 at 1:55