Cover
of a Topological Space| Cover
of a Topological Space |
In General
$ Def: For a
topological space (X, 
),
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 Ui's.
* Subcover: A subcollection of the Ui's 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 
(X, )
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};
((X,), )
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 
> 0
such that if diam(U) < ,
then
U some Ui
in the cover ( 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,)
of a compact subset K of a metric space X wrt > 0
is the smallest number of balls of radius that
will cover K.
@ References: Szarek m.FA/97, m.MG/97 [estimate
of N(K,)
for homogeneous spaces].
Related Topics
@ Coverage processes: Baccelli & Blaszczyszyn AAP(01)
[from Boolean model to Poisson–Voronoi tessellation and Johnson-Mehl
model].
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Send feedback and suggestions to bombelli at olemiss.edu – Modified
30 nov 2007