Definition df-kgen

Description: Define the "compact generator", the functor from topological spaces to compactly generated spaces. Also known as the k-ification operation. A set is k-open, i.e. x e. ( kGenj ) , iff the preimage of x is open in all compact Hausdorff spaces. (Contributed by Mario Carneiro, 20-Mar-2015)

	Ref	Expression
Assertion	df-kgen	⊢ 𝑘Gen = ( 𝑗 ∈ Top ↦ { 𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀ 𝑘 ∈ 𝒫 ∪ 𝑗 ( ( 𝑗 ↾t 𝑘 ) ∈ Comp → ( 𝑥 ∩ 𝑘 ) ∈ ( 𝑗 ↾t 𝑘 ) ) } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ckgen	⊢ 𝑘Gen
1		vj	⊢ 𝑗
2		ctop	⊢ Top
3		vx	⊢ 𝑥
4	1	cv	⊢ 𝑗
5	4	cuni	⊢ ∪ 𝑗
6	5	cpw	⊢ 𝒫 ∪ 𝑗
7		vk	⊢ 𝑘
8		crest	⊢ ↾t
9	7	cv	⊢ 𝑘
10	4 9 8	co	⊢ ( 𝑗 ↾t 𝑘 )
11		ccmp	⊢ Comp
12	10 11	wcel	⊢ ( 𝑗 ↾t 𝑘 ) ∈ Comp
13	3	cv	⊢ 𝑥
14	13 9	cin	⊢ ( 𝑥 ∩ 𝑘 )
15	14 10	wcel	⊢ ( 𝑥 ∩ 𝑘 ) ∈ ( 𝑗 ↾t 𝑘 )
16	12 15	wi	⊢ ( ( 𝑗 ↾t 𝑘 ) ∈ Comp → ( 𝑥 ∩ 𝑘 ) ∈ ( 𝑗 ↾t 𝑘 ) )
17	16 7 6	wral	⊢ ∀ 𝑘 ∈ 𝒫 ∪ 𝑗 ( ( 𝑗 ↾t 𝑘 ) ∈ Comp → ( 𝑥 ∩ 𝑘 ) ∈ ( 𝑗 ↾t 𝑘 ) )
18	17 3 6	crab	⊢ { 𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀ 𝑘 ∈ 𝒫 ∪ 𝑗 ( ( 𝑗 ↾t 𝑘 ) ∈ Comp → ( 𝑥 ∩ 𝑘 ) ∈ ( 𝑗 ↾t 𝑘 ) ) }
19	1 2 18	cmpt	⊢ ( 𝑗 ∈ Top ↦ { 𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀ 𝑘 ∈ 𝒫 ∪ 𝑗 ( ( 𝑗 ↾t 𝑘 ) ∈ Comp → ( 𝑥 ∩ 𝑘 ) ∈ ( 𝑗 ↾t 𝑘 ) ) } )
20	0 19	wceq	⊢ 𝑘Gen = ( 𝑗 ∈ Top ↦ { 𝑥 ∈ 𝒫 ∪ 𝑗 ∣ ∀ 𝑘 ∈ 𝒫 ∪ 𝑗 ( ( 𝑗 ↾t 𝑘 ) ∈ Comp → ( 𝑥 ∩ 𝑘 ) ∈ ( 𝑗 ↾t 𝑘 ) ) } )