Cover of a Topological Space  

In General
$ Def: For a topological space (X, τ), a collection {Ui} of (open) subsets of X whose union is X.
* Locally finite: For all xX, there is a neighborhood U which intersects finitely many Uis.
* Subcover: A subcollection of the Uis satisfying the same conditions.
* Set of covers of X: > see Quasiorder.
@ References: Isbell 64, p1.
> Related concepts: see paracompact topological space.

Operations on Covers; The Set of Covers of a Space
* Idea: For a given topological space X, the set \(\cal C\)(X, τ) of covers of X is a partially ordered (actually, directed) commutative semigroup, with the operations below.
* Refinement: Another covering {Vi}, such that (∀Vi, ∃ Uj such that Vi Uj).
* Meet: For two covers C and D, CD:= {UiVj | UiC, VjD}; (\(\cal C\)(X, τ), ∧) is a commutative semigroup.
* Star: For any cover C, C*:= {St(Ui, C) | UiC}, where for any AX we define St(A, C):= ∪UiA ≠ Ø U.

Of a Metric Space
* Uniform cover: One for which ∃ ε > 0 such that if diam(U) < ε, then U ⊂ some Ui in the cover (ε is a Lebesgue number for {Ui}).
* Lebesgue number: Every open cover of a metric space has a Lebesgue number.
* Covering number: The covering number N(K, ε) of a compact subset K of a metric space X with respect to ε > 0 is the smallest number of balls of radius ε that will cover K.
@ References: Szarek m.FA/97, m.MG/97 [estimate of N(K, ε) for homogeneous spaces].

Related Topics > s.a. measure theory [quantum covers].
* Nerve of a cover: Given a cover {Ui}iI of a space X, the nerve N is the abstract simplicial complex defined by the set of finite subsets of I such that: (i) The empty set belongs to N, and (ii) A finite set JI belongs to N if and only if the intersection of the Ui whose subindices are in J is non-empty; Another (similar, but not equivalent) definition is obtained by replacing the cover with the category whose objects are all intersections of elements of the original cover and whose morphisms are the inclusion relations, and then applying the categorical definition of nerve; A geometrical realization of the nerve may or may not be topologically equivalent to the original space; > s.a. Wikipedia page; Nerve; MathOverflow page; MathKB page.
@ Nerve of a cover: McCord PAMS(67) [homotopy type].
@ Coverage processes: Baccelli & Blaszczyszyn AAP(01) [from Boolean model to Poisson-Voronoi tessellation and Johnson-Mehl model].

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