On some spectral spaces associated to tensor triangulated categories

We consider a closure operator $c$ of finite type on the space $SMod(\mathcal M)$ of thick $\mathcal K$-submodules of a triangulated category $\mathcal M$ that is a module over a tensor triangulated category $(\mathcal K,\otimes,1)$. Our purpose is to show that the space $SMod^c(\mathcal M)$ of fixed points of the operator $c$ is a spectral space that also carries the structure of a topological monoid.