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Mirrors > Home > MPE Home > Th. List > df-t0 | Structured version Visualization version GIF version |
Description: Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2709): 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 22498) 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.) |
Ref | Expression |
---|---|
df-t0 | ⊢ Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ct0 22457 | . 2 class Kol2 | |
2 | vx | . . . . . . . . 9 setvar 𝑥 | |
3 | vo | . . . . . . . . 9 setvar 𝑜 | |
4 | 2, 3 | wel 2107 | . . . . . . . 8 wff 𝑥 ∈ 𝑜 |
5 | vy | . . . . . . . . 9 setvar 𝑦 | |
6 | 5, 3 | wel 2107 | . . . . . . . 8 wff 𝑦 ∈ 𝑜 |
7 | 4, 6 | wb 205 | . . . . . . 7 wff (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) |
8 | vj | . . . . . . . 8 setvar 𝑗 | |
9 | 8 | cv 1538 | . . . . . . 7 class 𝑗 |
10 | 7, 3, 9 | wral 3064 | . . . . . 6 wff ∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) |
11 | 2, 5 | weq 1966 | . . . . . 6 wff 𝑥 = 𝑦 |
12 | 10, 11 | wi 4 | . . . . 5 wff (∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) |
13 | 9 | cuni 4839 | . . . . 5 class ∪ 𝑗 |
14 | 12, 5, 13 | wral 3064 | . . . 4 wff ∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) |
15 | 14, 2, 13 | wral 3064 | . . 3 wff ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) |
16 | ctop 22042 | . . 3 class Top | |
17 | 15, 8, 16 | crab 3068 | . 2 class {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)} |
18 | 1, 17 | wceq 1539 | 1 wff Kol2 = {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)} |
Colors of variables: wff setvar class |
This definition is referenced by: ist0 22471 |
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