Non-trivial explicit example of a partition of unity

Question

Does exist a non-discrete paracompact example where is possible to give a partition of unity with the functions defined explicitly for a specific non trivial cover of the space?

Answer 1

Here is another simpler example that may help you visualize what partitions of unity are. Consider the open cover of $\mathbb{R}$ given by $U = (-\infty,2)$ and $V = (-2,\infty)$. A partition of unity associated to $\{U,V\}$ could be given by $\{ f_U, f_V \}$, where

$$ f_U, f_V \colon \mathbb{R} \to [0,1]$$

$$ f_U (x) = \left\{ \begin{array}{ll} 1 & \text{ if } x \leq -1 \\ \frac{1-x}{2} & \text{ if } -1 \leq x \leq 1 \\ 0 & \text{ if } x \geq 1 \end{array} \right. $$

$$ f_V (x) = \left\{ \begin{array}{ll} 0 & \text{ if } x \leq -1 \\ \frac{x-1}{2} & \text{ if } -1 \leq x \leq 1 \\ 1 & \text{ if } x \geq 1 \end{array} \right. $$

Then it is clear that $f_U(x)+f_V(x) = 1$ for all $x$. The support of $f_U$ is contained in $U$ and the support of $f_V$ in $V$. Of course the condition that any $x$ has an open neighbourhood where only a finite number of the functions are different from cero is automatic here because there are only two functions.

Answer 2

I do not know what your precise definition is of a partition of unity, but I like the example of $X=\mathbf R$, with open covering $\{\mathbf R\setminus \pi\mathbf Z,\mathbf R\setminus (\frac\pi2+\pi\mathbf Z)\}$, and functions $\sin^2$ and $\cos^2$.