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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 22491).
See istopg 22389 and istop2g 22390 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 22387 | . 2 class Top | |
2 | vy | . . . . . . . 8 setvar 𝑦 | |
3 | 2 | cv 1541 | . . . . . . 7 class 𝑦 |
4 | 3 | cuni 4908 | . . . . . 6 class ∪ 𝑦 |
5 | vx | . . . . . . 7 setvar 𝑥 | |
6 | 5 | cv 1541 | . . . . . 6 class 𝑥 |
7 | 4, 6 | wcel 2107 | . . . . 5 wff ∪ 𝑦 ∈ 𝑥 |
8 | 6 | cpw 4602 | . . . . 5 class 𝒫 𝑥 |
9 | 7, 2, 8 | wral 3062 | . . . 4 wff ∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 |
10 | vz | . . . . . . . . 9 setvar 𝑧 | |
11 | 10 | cv 1541 | . . . . . . . 8 class 𝑧 |
12 | 3, 11 | cin 3947 | . . . . . . 7 class (𝑦 ∩ 𝑧) |
13 | 12, 6 | wcel 2107 | . . . . . 6 wff (𝑦 ∩ 𝑧) ∈ 𝑥 |
14 | 13, 10, 6 | wral 3062 | . . . . 5 wff ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥 |
15 | 14, 2, 6 | wral 3062 | . . . 4 wff ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥 |
16 | 9, 15 | wa 397 | . . 3 wff (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥) |
17 | 16, 5 | cab 2710 | . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)} |
18 | 1, 17 | wceq 1542 | 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 22389 |
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