I am trying to wrap my head around on what partition of unity means and trying to understand some examples of it. I have an engineering background so most of the abstract discussions are a little hard for me to grasp without a clear explanation and examples. Also I have no formal education on topology, however I really want to self study this subject. The problem is due to a misunderstanding of some concepts so a clear explanation is appreciated.
On this page an example is given for a partition of unity on a circle $S^1$ as $\{sin^2(\theta/2),cos^2(\theta/2)\}$ subordinate to the covering $\{(0,2\pi),(-\pi,\pi)\}$.
The set of functions $\{sin^2(\theta/2),cos^2(\theta/2)\}$ add up to 1, so that condition is satisfied. However, if we denote the topological space by $X$ and the functions by $\epsilon_i$, according to the definition:
A partition of unity is subordinate to an open cover $\{U_i\}$ of $X$
if each $\epsilon_i$ is zero on the complement of $U_i$.
If we parameterize $S^1$ by an angle in the range $0 \leq \theta < 2\pi$, then it means that the function $\epsilon_1 = sin^2(\theta/2)$ must be zero on the complement of $U_1 = (0,2\pi)$. The comlpement is $U_1^c = 0$, hence $\epsilon_1$ vanishes on $U_1^c$. Also, the function $\epsilon_2 = cos^2(\theta/2)$ must be zero on the complement of $U_2 = (-\pi,\pi)$. First of all, what is the complement of $U_2$ in this context? Is it $U_2^c = [\pi,2\pi)$?! If it is, then there are plenty of points on $U_2^c$ that yield non-zero values for $\epsilon_2$! What am I doing wrong here?
Best Answer
The covering of $S^1$ that is given is misleading IMO.
The mean by $U_1 = (0,2\pi)$ the open set $U_1:= \{e^{ix}: x \in (0,2\pi)\}$, really. So they're referring to the unique angle in $[0,2\pi)$ that defines the point on $S^1$. Similarly for $U_2$.
If we look at $\sin^2(\frac\theta 2)$ (the value for $e^{i\theta} \in S^1$), we indeed see that is is $0$ iff $\frac\theta 2$ is an integer multiple of $\pi$, so iff $\theta$ is an integer multiple of $2\pi$, and so only $0$ when $\theta \notin U_1$ (recall that $\theta$ is chosen in $[0,2\pi)$ (main angle)).
Now reason similarly for $f(e^{i \theta})=\cos^2(\frac \theta 2)$: $0$ only for $\frac \theta 2$ of the form $\frac \pi 2 + k\pi, k \in \Bbb Z$ so $\theta$ of the form $\pi + k 2 \pi, k \in \Bbb Z$ and no such value lies in $U_2$.
Note that $U_2$ is $S^1$ minus the single point where the function is $0$ (i.e. $(-1,0)$) too, just like $U_1$ is.