I am reading Introduction to the Theory of Distributions by Friedlander and Joshi. As definition 8.6.1, they define the singular support of a tempered distribution $u$ to be the complement of {$x$: $u$ is $C^{\infty}$ on some neighborhood of $x$}.
From this definition I cannot see why the singular support of a tempered distribution need to be compact. But in proof for Lemma 8.6.1, they begin by choosing a test function $\phi$ such that $\phi\equiv 1$ on the singular support of a given tempered distribution.
Are they assuming that the singular support of a tempered distribution is compact? How can they do this?
Thanks!