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: More generally, if ''Y'' is a subset of ''X'' and ''C'' is a collection of subsets of ''X'' whose union contains ''Y'', then ''C'' is said to be a cover of ''Y''. 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 Set s (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 any ''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 Finite ly 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 : is finite. COMPACTNESS The language of covers is often used to define several topological properties related to ''compactness''. A topological space ''X'' is said to be
For some more variations see the above articles. SEE ALSO |
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