  Let $k$ be a non-Archimedean field, $X$ a $k$-affinoid space and $f$ an
analytic function over $X$. We precisely describe how the geometric connected
components of the spaces $\{x \in X, |f(x)| \ge \varepsilon\}$ behave with
regards to $\varepsilon$. We also obtain a result concerning privileged
neighbourhoods and adapt a theorem from complex analytic geometry about
Noetherianity for germs of analytic functions.
