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 $T$ 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 $V$.