Compactness is not hereditary

From Topospaces
Jump to: navigation, search
This article gives the statement, and possibly proof, of a topological space property (i.e., compact space) not satisfying a topological space metaproperty (i.e., subspace-hereditary property of topological spaces).
View all topological space metaproperty dissatisfactions | View all topological space metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for topological space properties
Get more facts about compact space|Get more facts about subspace-hereditary property of topological spaces|


Property-theoretic statement

The property of topological spaces of being a compact space is not a subspace-hereditary property of topological spaces.

Verbal statement

A subset of a compact space, equipped with the [[subspace topology], need not be a compact space.

Related facts


Consider the unit interval [0,1]. This is a closed and bounded subset of the real line, hence it is compact. The subset (0,1), equipped with the subspace topology, is not compact, because it has an open cover given by subsets of the form (1/(n+2),1/n) that has no finite subcover. (Alternatively, it is not compact because it is not a closed subset of the real line).