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Definition df-t0 21897
Description: Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2792): 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 21931) 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 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
Distinct variable group:   𝑗,𝑜,𝑥,𝑦

Detailed syntax breakdown of Definition df-t0
StepHypRef Expression
1 ct0 21890 . 2 class Kol2
2 vx . . . . . . . . 9 setvar 𝑥
3 vo . . . . . . . . 9 setvar 𝑜
42, 3wel 2115 . . . . . . . 8 wff 𝑥𝑜
5 vy . . . . . . . . 9 setvar 𝑦
65, 3wel 2115 . . . . . . . 8 wff 𝑦𝑜
74, 6wb 208 . . . . . . 7 wff (𝑥𝑜𝑦𝑜)
8 vj . . . . . . . 8 setvar 𝑗
98cv 1536 . . . . . . 7 class 𝑗
107, 3, 9wral 3125 . . . . . 6 wff 𝑜𝑗 (𝑥𝑜𝑦𝑜)
112, 5weq 1964 . . . . . 6 wff 𝑥 = 𝑦
1210, 11wi 4 . . . . 5 wff (∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)
139cuni 4814 . . . . 5 class 𝑗
1412, 5, 13wral 3125 . . . 4 wff 𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)
1514, 2, 13wral 3125 . . 3 wff 𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)
16 ctop 21477 . . 3 class Top
1715, 8, 16crab 3129 . 2 class {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
181, 17wceq 1537 1 wff Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 𝑗𝑦 𝑗(∀𝑜𝑗 (𝑥𝑜𝑦𝑜) → 𝑥 = 𝑦)}
Colors of variables: wff setvar class
This definition is referenced by:  ist0  21904
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