  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.
