We study the topological
μ
-calculus, based on both Cantor derivative and closure modalities, proving completeness, decidability and finite model property over general topological spaces, as well as over
T
0
and
T
D
spaces. We also investigate the relational
μ
-calculus, providing general completeness results for all natural fragments of the
μ
-calculus over many different classes of relational frames. Unlike most other such proofs for
μ
-calculi, ours is model-theoretic, making an innovative use of a known method from modal logic (–the ‘final’ submodel of the canonical model), that has the twin advantages of great generality and essential simplicity.