Theorem xkotop

Description: The compact-open topology is a topology. (Contributed by Mario Carneiro, 19-Mar-2015)

	Ref	Expression
Assertion	xkotop	⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ∈ Top ) → ( 𝑆 ↑ko 𝑅 ) ∈ Top )

Proof

Step	Hyp	Ref	Expression
1		eqid	⊢ ∪ 𝑅 = ∪ 𝑅
2		eqid	⊢ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } = { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp }
3		eqid	⊢ ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) = ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } )
4	1 2 3	xkoval	⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ∈ Top ) → ( 𝑆 ↑ko 𝑅 ) = ( topGen ‘ ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ) )
5		fibas	⊢ ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ∈ TopBases
6		tgcl	⊢ ( ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ∈ TopBases → ( topGen ‘ ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ) ∈ Top )
7	5 6	ax-mp	⊢ ( topGen ‘ ( fi ‘ ran ( 𝑘 ∈ { 𝑥 ∈ 𝒫 ∪ 𝑅 ∣ ( 𝑅 ↾t 𝑥 ) ∈ Comp } , 𝑣 ∈ 𝑆 ↦ { 𝑓 ∈ ( 𝑅 Cn 𝑆 ) ∣ ( 𝑓 “ 𝑘 ) ⊆ 𝑣 } ) ) ) ∈ Top
8	4 7	eqeltrdi	⊢ ( ( 𝑅 ∈ Top ∧ 𝑆 ∈ Top ) → ( 𝑆 ↑ko 𝑅 ) ∈ Top )