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Mirrors > Home > ILE Home > Th. List > df-top | GIF version |
Description: Define the class of
topologies. It is a proper class. See istopg 12791 and
istopfin 12792 for the corresponding characterizations,
using respectively
binary intersections like in this definition and nonempty finite
intersections.
The final form of the definition is due to Bourbaki (Def. 1 of [BourbakiTop1] p. I.1), while the idea of defining a topology in terms of its open sets is due to Aleksandrov. For the convoluted history of the definitions of these notions, see Gregory H. Moore, The emergence of open sets, closed sets, and limit points in analysis and topology, Historia Mathematica 35 (2008) 220--241. (Contributed by NM, 3-Mar-2006.) (Revised by BJ, 20-Oct-2018.) |
Ref | Expression |
---|---|
df-top | ⊢ Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctop 12789 | . 2 class Top | |
2 | vy | . . . . . . . 8 setvar 𝑦 | |
3 | 2 | cv 1347 | . . . . . . 7 class 𝑦 |
4 | 3 | cuni 3796 | . . . . . 6 class ∪ 𝑦 |
5 | vx | . . . . . . 7 setvar 𝑥 | |
6 | 5 | cv 1347 | . . . . . 6 class 𝑥 |
7 | 4, 6 | wcel 2141 | . . . . 5 wff ∪ 𝑦 ∈ 𝑥 |
8 | 6 | cpw 3566 | . . . . 5 class 𝒫 𝑥 |
9 | 7, 2, 8 | wral 2448 | . . . 4 wff ∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 |
10 | vz | . . . . . . . . 9 setvar 𝑧 | |
11 | 10 | cv 1347 | . . . . . . . 8 class 𝑧 |
12 | 3, 11 | cin 3120 | . . . . . . 7 class (𝑦 ∩ 𝑧) |
13 | 12, 6 | wcel 2141 | . . . . . 6 wff (𝑦 ∩ 𝑧) ∈ 𝑥 |
14 | 13, 10, 6 | wral 2448 | . . . . 5 wff ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥 |
15 | 14, 2, 6 | wral 2448 | . . . 4 wff ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥 |
16 | 9, 15 | wa 103 | . . 3 wff (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥) |
17 | 16, 5 | cab 2156 | . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} |
18 | 1, 17 | wceq 1348 | 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} |
Colors of variables: wff set class |
This definition is referenced by: istopg 12791 |
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