In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano (1817) … See more

Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite See more

• A compact subset of a Hausdorff space X is closed.
• In any topological vector space (TVS), a compact subset is complete. However, every non-Hausdorff … See more

• Compactly generated space
• Compactness theorem
• Eberlein compactum
• Exhaustion by compact sets
• Lindelöf space See more

Any finite space is compact; a finite subcover can be obtained by selecting, for each point, an open set containing it. A nontrivial example of a compact space is the (closed) unit interval [0,1] of real numbers. If one chooses an infinite number of distinct … See more

• A closed subset of a compact space is compact.
• A finite union of compact sets is compact. See more

• Any finite topological space, including the empty set, is compact. More generally, any space with a finite topology (only finitely many open sets) is compact; this includes in particular the trivial topology.
• Any space carrying the cofinite topology is compact. See more