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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 2803): 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 21529) 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 21488 | . 2 class Kol2 | |
2 | vx | . . . . . . . . 9 setvar 𝑥 | |
3 | vo | . . . . . . . . 9 setvar 𝑜 | |
4 | 2, 3 | wel 2165 | . . . . . . . 8 wff 𝑥 ∈ 𝑜 |
5 | vy | . . . . . . . . 9 setvar 𝑦 | |
6 | 5, 3 | wel 2165 | . . . . . . . 8 wff 𝑦 ∈ 𝑜 |
7 | 4, 6 | wb 198 | . . . . . . 7 wff (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) |
8 | vj | . . . . . . . 8 setvar 𝑗 | |
9 | 8 | cv 1655 | . . . . . . 7 class 𝑗 |
10 | 7, 3, 9 | wral 3117 | . . . . . 6 wff ∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) |
11 | 2, 5 | weq 2061 | . . . . . 6 wff 𝑥 = 𝑦 |
12 | 10, 11 | wi 4 | . . . . 5 wff (∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) |
13 | 9 | cuni 4660 | . . . . 5 class ∪ 𝑗 |
14 | 12, 5, 13 | wral 3117 | . . . 4 wff ∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) |
15 | 14, 2, 13 | wral 3117 | . . 3 wff ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) |
16 | ctop 21075 | . . 3 class Top | |
17 | 15, 8, 16 | crab 3121 | . 2 class {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)} |
18 | 1, 17 | wceq 1656 | 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 21502 |
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