Definition:Join of Open Covers

Definition

Let $S$ be a topological space.

Let $\alpha, \beta$ be open covers for $S$.

The join of $\alpha$ and $\beta$ is the open cover:

$\alpha \vee \beta := \set {A \cap B : A \in \alpha, B \in \beta}$

Similarly, given open covers $\alpha_1, \ldots, \alpha_n$ for $S$:

$\ds \bigvee_{k \mathop = 1}^n \alpha_k := \set { \bigcap_{k \mathop = 1}^n A_k \; : A_k \in \alpha_k \; \text{for}\; \forall k }$

Also see

• Definition:Join of Finite Partitions

Sources

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