Definition:Even Cover

Definition

Let $T = \struct {S, \tau}$ be a topological space.

Let $\UU$ be a cover of $S$.

Let $S \times S$ denote the cartesian product of $S$ with itself.

Let $\tau_{S \times S}$ denote the product topology on $S \times S$.

Let $T \times T$ denote the product space $\struct {S \times S, \tau_{S \times S} }$.

Then $\UU$ is an even cover of $T$ if and only if there exists a neighborhood $V$ of the diagonal $\Delta_S$ of $S \times S$ in $T \times T$:

$\set{\map V x : x \in S}$ is a refinement of $\UU$

where:

$V$ is seen as a relation on $S \times S$

$\map V x$ denotes the image of $x$ under $V$.

Sources

• 1955: John L. Kelley: General Topology: Chapter $5$: Compact Spaces, $\S$ Lebesgue's Covering Lemma