Definition:Even Cover
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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:
Sources
- 1955: John L. Kelley: General Topology: Chapter $5$: Compact Spaces, $\S$ Lebesgue's Covering Lemma