COVER (TOPOLOGY)

(Redirected from Open cover)
In mathematics, a 'cover' of a set ''X'' is a collection of sets ''C'' whose union is ''X''. In symbols, if ''C'' = {''U''_α : α ∈ ''A''} is an indexed family of subsets ''U'', of ''X'', then ''C'' is a cover if
:
More generally, if ''Y'' is a subset of ''X'', and ''C'' is a collection of subsets ''U''_α of ''X'', whose union contains ''Y'', then ''C'' is said to be a cover of ''Y''. i.e. ''C'' is a cover of ''Y'' if
:
Covers are commonly used in the context of topology. If the set ''X'' is a topological space, we say that ''C'' is an 'open cover' if each of its members are open sets (i.e. each ''U''_α is contained in ''T'', where ''T'' is the topology on ''X'').
If ''C'' is a cover of ''X'' then a 'subcover' of ''C'' is a subset of ''C'' which still covers ''X''.
A 'refinement' of a cover ''C'' of ''X'' is a new cover ''D'' of ''X'' such that every set in ''D'' is contained in some set in ''C''. In symbols, the cover ''D'' = {''V''_β : β ∈ ''B''} is a refinement of the cover ''C'' = {''U''_α : α ∈ ''A''} if for every ''V''_β there exists some ''U''_α such that ''V''_β ⊆ ''U''_α.
Every subcover is also a refinement, but not vice-versa. Note however that a refinement will, in general, have ''more'' sets than the original cover.
An open cover of ''X'' is said to be 'locally finite' if every point of ''X'' has a neighborhood which intersects only finitely many sets in the cover. In symbols, ''C'' = {''U''_α} is locally finite if for any ''x'' ∈ ''X'', there exists some neighborhood ''N''(''x'') of ''x'' such that the set
:
eq arnothing
ight}
is finite.

Contents

Compactness

See also

Compactness

The language of covers is often used to define several topological properties related to ''compactness''. A topological space ''X'' is said to be

★ ''compact'' if every open cover has a finite subcover.

★ ''Lindelöf'' if every open cover has a countable subcover.

★ ''metacompact'' if every open cover has a point finite open refinement.

★ ''paracompact'' if every open cover admits a locally finite, open refinement.
For some more variations see the above articles.

See also

★ Covering space

★ Atlas (topology)

★ Lebesgue covering dimension