Cover of a Topological Space| 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 x ∈ X,
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, C ∧ D:=
{Ui ∩ Vj | Ui ∈ C, Vj ∈ D};
(\(\cal C\)(X, τ), ∧)
is a commutative semigroup.
* Star: For any cover C,
C*:= {St(Ui, C)
| Ui ∈ C},
where for any A ⊂ X we define St(A, C):=
∪Ui ∩ A ≠
Ø Ui .
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}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 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].
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 16 jan 2016