Definition df-t0 21117

Description: Define T0 or Kolmogorov spaces. A T₀ space satisfies a kind of "topological extensionality" principle (compare ax-ext 2602): any two points which are members of the same open sets are equal, or in contraposition, for any two distinct points there is an open set which contains one point but not the other. This differs from T₁ spaces (see ist1-2 21151) in that in a T₁ space you can choose which point will be in the open set and which outside; in a T₀ space you only know that one of the two points is in the set. (Contributed by Jeff Hankins, 1-Feb-2010.)

Assertion

Ref	Expression
df-t0	⊢ Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)}

Distinct variable group:   𝑗,𝑜,𝑥,𝑦

Detailed syntax breakdown of Definition df-t0

Step	Hyp	Ref	Expression
1		ct0 21110	. 2 class Kol2
2		vx	. . . . . . . . 9 setvar 𝑥
3		vo	. . . . . . . . 9 setvar 𝑜
4	2, 3	wel 1991	. . . . . . . 8 wff 𝑥 ∈ 𝑜
5		vy	. . . . . . . . 9 setvar 𝑦
6	5, 3	wel 1991	. . . . . . . 8 wff 𝑦 ∈ 𝑜
7	4, 6	wb 196	. . . . . . 7 wff (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)
8		vj	. . . . . . . 8 setvar 𝑗
9	8	cv 1482	. . . . . . 7 class 𝑗
10	7, 3, 9	wral 2912	. . . . . 6 wff ∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜)
11	2, 5	weq 1874	. . . . . 6 wff 𝑥 = 𝑦
12	10, 11	wi 4	. . . . 5 wff (∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)
13	9	cuni 4436	. . . . 5 class ∪ 𝑗
14	12, 5, 13	wral 2912	. . . 4 wff ∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)
15	14, 2, 13	wral 2912	. . 3 wff ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)
16		ctop 20698	. . 3 class Top
17	15, 8, 16	crab 2916	. 2 class {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)}
18	1, 17	wceq 1483	1 wff Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)}

This definition is referenced by:  ist0  21124