Characterization of Zero Entropy of Open Cover

Theorem

Let $X$ be a topological Space.

Let $\alpha$ be an open cover of $X$.

Let $\map H \alpha$ be the entropy of $\alpha$.

Then $\map H \alpha = 0$ if and only if $X \in \alpha$.

Proof

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Sources

• 2013: Peter Walters: An Introduction to Ergodic Theory (4th ed.): Chapter $7$: Topological Entropy