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Mirrors > Home > MPE Home > Th. List > df-top | Structured version Visualization version GIF version |
Description: Define the class of
topologies. It is a proper class (see topnex 21276).
See istopg 21175 and istop2g 21176 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 21173 | . 2 class Top | |
2 | vy | . . . . . . . 8 setvar 𝑦 | |
3 | 2 | cv 1519 | . . . . . . 7 class 𝑦 |
4 | 3 | cuni 4739 | . . . . . 6 class ∪ 𝑦 |
5 | vx | . . . . . . 7 setvar 𝑥 | |
6 | 5 | cv 1519 | . . . . . 6 class 𝑥 |
7 | 4, 6 | wcel 2079 | . . . . 5 wff ∪ 𝑦 ∈ 𝑥 |
8 | 6 | cpw 4447 | . . . . 5 class 𝒫 𝑥 |
9 | 7, 2, 8 | wral 3103 | . . . 4 wff ∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 |
10 | vz | . . . . . . . . 9 setvar 𝑧 | |
11 | 10 | cv 1519 | . . . . . . . 8 class 𝑧 |
12 | 3, 11 | cin 3853 | . . . . . . 7 class (𝑦 ∩ 𝑧) |
13 | 12, 6 | wcel 2079 | . . . . . 6 wff (𝑦 ∩ 𝑧) ∈ 𝑥 |
14 | 13, 10, 6 | wral 3103 | . . . . 5 wff ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥 |
15 | 14, 2, 6 | wral 3103 | . . . 4 wff ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥 |
16 | 9, 15 | wa 396 | . . 3 wff (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥) |
17 | 16, 5 | cab 2773 | . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} |
18 | 1, 17 | wceq 1520 | 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} |
Colors of variables: wff setvar class |
This definition is referenced by: istopg 21175 |
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