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Definition df-top 20747
Description: Define the class of topologies. It is a proper class (see topnex 20848). See istopg 20748 and istop2g 20749 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.)

Assertion
Ref Expression
df-top Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-top
StepHypRef Expression
1 ctop 20746 . 2 class Top
2 vy . . . . . . . 8 setvar 𝑦
32cv 1522 . . . . . . 7 class 𝑦
43cuni 4468 . . . . . 6 class 𝑦
5 vx . . . . . . 7 setvar 𝑥
65cv 1522 . . . . . 6 class 𝑥
74, 6wcel 2030 . . . . 5 wff 𝑦𝑥
86cpw 4191 . . . . 5 class 𝒫 𝑥
97, 2, 8wral 2941 . . . 4 wff 𝑦 ∈ 𝒫 𝑥 𝑦𝑥
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1522 . . . . . . . 8 class 𝑧
123, 11cin 3606 . . . . . . 7 class (𝑦𝑧)
1312, 6wcel 2030 . . . . . 6 wff (𝑦𝑧) ∈ 𝑥
1413, 10, 6wral 2941 . . . . 5 wff 𝑧𝑥 (𝑦𝑧) ∈ 𝑥
1514, 2, 6wral 2941 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥
169, 15wa 383 . . 3 wff (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)
1716, 5cab 2637 . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
181, 17wceq 1523 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  istopg  20748
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