TITLE: Products of Manifolds
QUESTION [5 upvotes]: Let $M$ and $K$ be manifolds, and let $(f_{\alpha})_{\alpha \in I}$ and $(h_{\gamma})_{\gamma \in J}$ be partitions unity for $M$ and $K$ respectively. Show that the collection of mappings on $M \times K$ given by all products $f_{\alpha}(a)h_{\gamma}(b)$ is a partition of unity for $M \times N$.  
I've fooled around with this and still I can't find the idea on how to show this. Any hints and help would do. Thank you very much!!

REPLY [5 votes]: Fix $(m,k)\in M\times K$. Then there are neighbourhoods $M_m\subseteq M$ and $K_k\subseteq K$ of $m$ and $k$ such that finitely many $f_\alpha$ and $h_\gamma$ are nonzero, respectively. Take a look at $M_m\times K_k$.
Lastly, note that $$\sum_{\alpha\in I}\sum_{\gamma\in J} f_\alpha(m)h_\gamma(k)=\left(\sum_{\alpha\in I} f_\alpha(m)\right)\left(\sum_{\gamma\in J} h_\gamma(k)\right).$$