Definition:Entropy of Open Cover

Definition

Let $X$ be a topological Space.

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

The entropy of $\alpha$ is defined as:

$\map H \alpha := \ln \map N \alpha$

where:

$\ln$ denotes the natural logarithm

$\map N \alpha := \min \leftset {\card \beta : \beta}$ is a subcover of $\rightset \alpha$

$\card \beta$ denotes the cardinality of $\beta$

Sources

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