Metamath Proof Explorer

Theorem kgentop

Description: A compactly generated space is a topology. (Note: henceforth we will use the idiom " J e. ran kGen " to denote " J is compactly generated", since as we will show a space is compactly generated iff it is in the range of the compact generator.) (Contributed by Mario Carneiro, 20-Mar-2015)

Ref Expression
Assertion kgentop J ran 𝑘Gen J Top


Step Hyp Ref Expression
1 kgenf 𝑘Gen : Top Top
2 frn 𝑘Gen : Top Top ran 𝑘Gen Top
3 1 2 ax-mp ran 𝑘Gen Top
4 3 sseli J ran 𝑘Gen J Top