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

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
compact space	every open cover has a finite subcover	compact implies compactly generated	compactly generated not implies compact	|FULL LIST, MORE INFO
locally compact space	every point is contained in an open subset contained in a closed, compact subset	locally compact implies compactly generated	compactly generated not implies locally compact	|FULL LIST, MORE INFO
first-countable space	countable basis at every point	first-countable implies compactly generated	compactly generated not implies first-countable	|FULL LIST, MORE INFO
metrizable space	underlying topology of a metric space	metrizable implies compactly generated	compactly generated not implies metrizable	|FULL LIST, MORE INFO
CW-space	underlying topology of a CW-complex	CW implies compactly generated	compactly generated not implies CW	|FULL LIST, MORE INFO
compactly generated Hausdorff space	compactly generated and Hausdorff			|FULL LIST, MORE INFO

References

Textbook references

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