Topology MOC

Cover

Let be a set A collection of subsets of is called a cover iff . topology Typically is a topological space, in which case is called an open cover iff every is open.

Further terminology

• A subcover of is a a subcollection of that is also a cover of .
• A refinement of is a cover such that every is contained in at least one , i.e. .
• A cover is locally finite iff every as a neighbourhood intersecting with finitely many .

Properties

• A space is compact iff every open cover has a finite subcover.
• A space is paracompact iff every open cover has a locally finite open refinement.

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