Definition df-kq 23703

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 23702	. 2 class KQ
2		vj	. . 3 setvar 𝑗
3		ctop 22900	. . 3 class Top
4	2	cv 1538	. . . 4 class 𝑗
5		vx	. . . . 5 setvar 𝑥
6	4	cuni 4906	. . . . 5 class ∪ 𝑗
7		vy	. . . . . . 7 setvar 𝑦
8	5, 7	wel 2108	. . . . . 6 wff 𝑥 ∈ 𝑦
9	8, 7, 4	crab 3435	. . . . 5 class {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}
10	5, 6, 9	cmpt 5224	. . . 4 class (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})
11		cqtop 17549	. . . 4 class qTop
12	4, 10, 11	co 7432	. . 3 class (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦}))
13	2, 3, 12	cmpt 5224	. 2 class (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))
14	1, 13	wceq 1539	1 wff KQ = (𝑗 ∈ Top ↦ (𝑗 qTop (𝑥 ∈ ∪ 𝑗 ↦ {𝑦 ∈ 𝑗 ∣ 𝑥 ∈ 𝑦})))

This definition is referenced by:  kqval  23735  kqtop  23754  kqf  23756