Finite Type Points On Subsets Of ${\mathbb{C}}^n$

In [4], D’Angelo introduced the notion of points of finite type for a real hypersurface $M$ in ${\mathbb{C}}^n$ and showed that the set of points of finite type in $M$ is open. Later, Lamel-Mir [8] considered a natural extension of D’Angelo’s definition for an arbitrary set $M$ in ${\mathbb{C}}^n$. Building on D’Angelo’s work, we prove the openness of the set of points of finite type for any subset $M$ in ${\mathbb{C}}^n\mathrm{.}$