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For a smooth stationary Gaussian field on $\mathbb{R}^d$ and level $\ell \in\mathbb{R}$, we consider the number of connected components of the excursionset $\{f \ge \ell\}$ (or level set $\{f = \ell\}$) contained in large domains.The mean of this quantity is known to scale like the volume of the domain undergeneral assumptions on the field. We prove that, assuming sufficient decay ofcorrelations (e.g. the Bargmann-Fock field), a central limit theorem holds withvolume-order scaling. Previously such a result had only been established for`additive' geometric functionals of the excursion/level sets (e.g. the volumeor Euler characteristic) using Hermite expansions. Our approach, based on amartingale analysis, is more robust and can be generalised to a wider class oftopological functionals. A major ingredient in the proof is a third momentbound on critical points, which is of independent interest.

A central limit theorem for the number of excursion set components of Gaussian fields