Let $(V_{i})_{i\in I}$ be an open covering of an open subset $V$ of ${X^{\rm an}}$. A smooth partition of unity on $V$ with compact supports subordinated to the covering $(V_{i})_{i\in I}$ is a family $(\phi_{j})_{j\in J}$ of non-negative smooth functions with compact support on $V$ with the following properties:

• (i)

  The family $({\rm supp}(\phi_{j}))_{j\in J}$ is locally finite on $V$.

• (ii)

  We have $\sum_{j\in J}\phi_{j}\equiv 1$ on $V$.

• (iii)

  For every $j\in J$, there is $i(j)\in I$ such that ${\rm supp}(\phi_{j})\subset V_{i(j)}$.