Compactly generated space

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 $$i \in I$$.

Textbook references

 * , Page 283 (formal definition)