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

Symbol-free 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.

Definition with symbols

A topological space $X$ is said to be compactly generated if there exists a collection $\{K_i{\}}_{i\in I}$ of compact subsets of $X$, such that a subset $U\subset X$ is open if and only if $U\cap K_i$ is open in $K_i$ for every $i\in I$.

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)