Definition df-kq 23636

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 23635	. 2 class KQ
2		vj	. . 3 setvar 𝑗
3		ctop 22835	. . 3 class Top
4	2	cv 1540	. . . 4 class 𝑗
5		vx	. . . . 5 setvar 𝑥
6	4	cuni 4861	. . . . 5 class ∪ 𝑗
7		vy	. . . . . . 7 setvar 𝑦
8	5, 7	wel 2114	. . . . . 6 wff 𝑥 ∈ 𝑦
9	8, 7, 4	crab 3397	. . . . 5 class {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}
10	5, 6, 9	cmpt 5177	. . . 4 class (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})
11		cqtop 17422	. . . 4 class qTop
12	4, 10, 11	co 7356	. . 3 class (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))
13	2, 3, 12	cmpt 5177	. 2 class (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
14	1, 13	wceq 1541	1 wff KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))

This definition is referenced by:  kqval  23668  kqtop  23687  kqf  23689