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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 an ordered pair where is a set and is a topology on , meaning:
Here denotes the power set of , and the members of are the open sets (whose complements are the closed sets). Many standard constructions—such as the subspace topology, product topology, and quotient topology—produce new topological spaces from existing ones.
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 such that its complement (the set difference) is a open set.
Closed sets are the natural targets of the closure operation: the closure of is the smallest closed set containing . Closed sets also provide an equivalent formulation of continuity via preimages of closed sets.
The subspace topology on a subset of a topological space is the topology
With this topology, becomes a topological space, called a subspace of .
A subset is open in the subspace exactly when it is the intersection of with an open set of . The inclusion map is automatically continuous for the subspace topology.
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.”