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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 2590): 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 20961) 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 20920 | . 2 class Kol2 | |
2 | vx | . . . . . . . . 9 setvar 𝑥 | |
3 | vo | . . . . . . . . 9 setvar 𝑜 | |
4 | 2, 3 | wel 1978 | . . . . . . . 8 wff 𝑥 ∈ 𝑜 |
5 | vy | . . . . . . . . 9 setvar 𝑦 | |
6 | 5, 3 | wel 1978 | . . . . . . . 8 wff 𝑦 ∈ 𝑜 |
7 | 4, 6 | wb 195 | . . . . . . 7 wff (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) |
8 | vj | . . . . . . . 8 setvar 𝑗 | |
9 | 8 | cv 1474 | . . . . . . 7 class 𝑗 |
10 | 7, 3, 9 | wral 2896 | . . . . . 6 wff ∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) |
11 | 2, 5 | weq 1861 | . . . . . 6 wff 𝑥 = 𝑦 |
12 | 10, 11 | wi 4 | . . . . 5 wff (∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) |
13 | 9 | cuni 4372 | . . . . 5 class ∪ 𝑗 |
14 | 12, 5, 13 | wral 2896 | . . . 4 wff ∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) |
15 | 14, 2, 13 | wral 2896 | . . 3 wff ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦) |
16 | ctop 20517 | . . 3 class Top | |
17 | 15, 8, 16 | crab 2900 | . 2 class {𝑗 ∈ Top ∣ ∀𝑥 ∈ ∪ 𝑗∀𝑦 ∈ ∪ 𝑗(∀𝑜 ∈ 𝑗 (𝑥 ∈ 𝑜 ↔ 𝑦 ∈ 𝑜) → 𝑥 = 𝑦)} |
18 | 1, 17 | wceq 1475 | 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 20934 |
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