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A closed subset of a compact set is compact
Closed subset of compact set is compact: Let be a topological space and let be compact. If is closed in the subspace topology on (equivalently, if for some closed ), then is compact.
This permanence property is frequently combined with the finite intersection property characterization of compactness.
A topological space is a pair , where is a set and is a collection of subsets of , satisfying:
The collection is the topology of the space.
A compact set is a subset of a topological space such that every open cover of contains a finite subcover: whenever is a family of open sets with , there exist indices such that
A closed set in a topological space is a subset whose complement is a open set.
The subspace topology on a subset of a topological space is the topology
With this topology, becomes a topological space, called a subspace of .
Finite intersection property theorem: Let be a topological space and let be given the subspace topology. Then is compact if and only if for every family of closed subsets of with the finite intersection property, one has .
This is an alternative to the open cover definition of compactness and is often the most convenient form for “compactness by contradiction.”