Cover of a Topological Space |

**In General**

$ __Def__: For a
topological space (*X*, *τ*),
a collection {*U*_{i}} of (open)
subsets of *X* whose union is *X*.

* __Locally finite__: For all *x* ∈ *X*,
there is a neighborhood *U* which
intersects finitely many *U*_{i}s.

* __Subcover__: A subcollection
of the *U*_{i}s satisfying the same conditions.

* __Set of covers of X__: > see Quasiorder.

@

>

**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 {*V*_{i}}, such that
(∀*V*_{i},
∃ *U*_{j}
such that *V*_{i }⊂ *U*_{j}).

* __Meet__: For two covers *C* and *D*, *C* ∧ *D*:=
{*U*_{i} ∩ *V*_{j} | *U*_{i} ∈ *C*, *V*_{j} ∈ *D*};
(\(\cal C\)(*X*,* τ*), ∧)
is a commutative semigroup.

* __Star__: For any cover *C*, *C**:=
{St(*U*_{i},* C*)
| *U*_{i} ∈ *C*},
where for any *A* ⊂ *X* we
define St(*A*,* C*):= ∪_{Ui
∩ A ≠ Ø} *U*_{i }.

**Of a Metric Space**

* __Uniform cover__: One
for which ∃ *ε* > 0
such that if diam(*U*) < *ε*, then
*U* ⊂ some *U*_{i}
in the cover (*ε* is
a Lebesgue number for {*U*_{i}}).

* __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 {*U*_{i}}_{i ∈ I} 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 *J* ⊆ *I* belongs
to *N* if and only if the intersection of the *U*_{i} 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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jan 2016