Compactly generated space

This article defines a property of topological spaces: a property that can be evaluated to true/false for any topological space|View a complete list of properties of topological spaces

This is a variation of compactness. View other variations of compactness

Definition

A topological space is said to be compactly generated if the topology on it is generated by a collection of compact subsets. In other words, a set in the topological space is open if and only if its intersection with each of the compact subsets is open, in the subspace topology.

Relation with other properties

Stronger properties

• Compact space
• Locally compact space
• First-countable space: For full proof, refer: First-countable implies compactly generated
• Metrizable space
• CW-space

References

Textbook references

• Topology (2nd edition) by James R. Munkres^{More info}, Page 283 (formal definition)