Definition df-kq 23659

Description: Define the Kolmogorov quotient. This is a function on topologies which maps a topology to its quotient under the topological distinguishability map, which takes a point to the set of open sets that contain it. Two points are mapped to the same image under this function iff they are topologically indistinguishable. (Contributed by Mario Carneiro, 25-Aug-2015.)

Assertion

Ref	Expression
df-kq	⊢ KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))

Distinct variable group:   𝑥,𝑗,𝑦

Detailed syntax breakdown of Definition df-kq

Step	Hyp	Ref	Expression
1		ckq 23658	. 2 class KQ
2		vj	. . 3 setvar 𝑗
3		ctop 22858	. . 3 class Top
4	2	cv 1541	. . . 4 class 𝑗
5		vx	. . . . 5 setvar 𝑥
6	4	cuni 4850	. . . . 5 class ∪ 𝑗
7		vy	. . . . . . 7 setvar 𝑦
8	5, 7	wel 2115	. . . . . 6 wff 𝑥 ∈ 𝑦
9	8, 7, 4	crab 3389	. . . . 5 class {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}
10	5, 6, 9	cmpt 5166	. . . 4 class (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})
11		cqtop 17467	. . . 4 class qTop
12	4, 10, 11	co 7367	. . 3 class (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))
13	2, 3, 12	cmpt 5166	. 2 class (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
14	1, 13	wceq 1542	1 wff KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))

This definition is referenced by:  kqval  23691  kqtop  23710  kqf  23712