Local boundedness of Catlin q-type

In [6], D’Angelo introduced the notion of finite type for points $p$ of a real hypersurface $M$ of ${\mathbb{C}}^n$ by defining the order of contact ${\mathrm{\Delta }}_q(M,p)$ of complex analytic $q$-dimensional varieties with $M$ at $p$. Later, Catlin [4] defined $q$-type, $D_q(M,p)$ for points of hypersurfaces by considering generic $(n-q+1)$-dimensional complex affine subspaces of $\mathbb C^n$\mathrm{.}WedefineageneralizationoftheCatli{n}^{\mathrm{\prime }}s$q$-type for an arbitrary subset $M$ of ${\mathbb{C}}^n$ in a similar way that D’Angelo’s 1-type, ${\mathrm{\Delta }}_1(M,p)$, is generalized in [13]. Using recent results connecting the D’Angelo and Catlin $q$-types in [1] and building on D’Angelo’s work on the openness of the set of points of finite ${\mathrm{\Delta }}_q$-type, we prove the openness of the set of points of finite Catlin $q$-type for an arbitrary subset $M\subset {\mathbb{C}}^n$.