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On a smooth, bounded pseudoconvex domain $\Omega$ in $\mathbb{C}^n$, toverify that Catlin's Property ($P$) holds for $b\Omega$, it suffices to checkthat it holds on the set of D'Angelo infinite type boundary points. In thisnote, we consider the support of the Levi core,$S_{\mathfrak{C}(\mathcal{N})}$, a subset of the infinite type points, and showthat Property ($P$) holds for $b\Omega$ if and only if it holds for$S_{\mathfrak{C}(\mathcal{N})}$. Consequently, if Property ($P$) holds on$S_{\mathfrak{C}(\mathcal{N})}$, then the $\overline{\partial}$-Neumannoperator $N_1$ is compact on $\Omega$.

Sufficient condition for compactness of the $\overline{\partial}$-Neumann operator using the Levi core