Theorem kgenftop

Description: The compact generator generates a topology. (Contributed by Mario Carneiro, 20-Mar-2015)

	Ref	Expression
Assertion	kgenftop	⊢ ( 𝐽 ∈ Top → ( 𝑘Gen ‘ 𝐽 ) ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		toptopon2	⊢ ( 𝐽 ∈ Top ↔ 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) )
2		kgentopon	⊢ ( 𝐽 ∈ ( TopOn ‘ ∪ 𝐽 ) → ( 𝑘Gen ‘ 𝐽 ) ∈ ( TopOn ‘ ∪ 𝐽 ) )
3	1 2	sylbi	⊢ ( 𝐽 ∈ Top → ( 𝑘Gen ‘ 𝐽 ) ∈ ( TopOn ‘ ∪ 𝐽 ) )
4		topontop	⊢ ( ( 𝑘Gen ‘ 𝐽 ) ∈ ( TopOn ‘ ∪ 𝐽 ) → ( 𝑘Gen ‘ 𝐽 ) ∈ Top )
5	3 4	syl	⊢ ( 𝐽 ∈ Top → ( 𝑘Gen ‘ 𝐽 ) ∈ Top )