Spatial logic of modal mu-calculus and tangled closure operators

There has been renewed interest in recent years in McKinsey and Tarski's interpretation of modal logic in topological spaces and their proof that S4 is the logic of any separable dense-in-itself metric space. Here we extend this work to the modal mu-calculus and to a logic of tangled closure operators that was developed by Fern\'andez-Duque after these two languages had been shown by Dawar and Otto to have the same expressive power over finite transitive Kripke models. We prove that this equivalence remains true over topological spaces. We establish the finite model property in Kripke semantics for various tangled closure logics with and without the universal modality $\forall$. We also extend the McKinsey--Tarski topological `dissection lemma'. These results are used to construct a representation map (also called a d-p-morphism) from any dense-in-itself metric space $X$ onto any finite connected locally connected serial transitive Kripke frame. This yields completeness theorems over $X$ for a number of languages: (i) the modal mu-calculus with the closure operator $\Diamond$; (ii) $\Diamond$ and the tangled closure operators $\langle t \rangle$; (iii) $\Diamond,\forall$; (iv) $\Diamond,\forall,\langle t \rangle$; (v) the derivative operator $\langle d \rangle$; (vi) $\langle d \rangle$ and the associated tangled closure operators $\langle dt \rangle$; (vii) $\langle d \rangle,\forall$; (viii) $\langle d \rangle,\forall,\langle dt \rangle$. Soundness also holds, if: (a) for languages with $\forall$, $X$ is connected; and (b) for languages with $\langle d \rangle$, $X$ validates the well known axiom $\mathrm{G}_1$. For countable languages without $\forall$, we prove strong completeness. We also show that in the presence of $\forall$, strong completeness fails if $X$ is compact and locally connected.