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Definition df-top 22747
Description: Define the class of topologies. It is a proper class (see topnex 22850). See istopg 22748 and istop2g 22749 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 22746 . 2 class Top
2 vy . . . . . . . 8 setvar 𝑦
32cv 1532 . . . . . . 7 class 𝑦
43cuni 4902 . . . . . 6 class 𝑦
5 vx . . . . . . 7 setvar 𝑥
65cv 1532 . . . . . 6 class 𝑥
74, 6wcel 2098 . . . . 5 wff 𝑦𝑥
86cpw 4597 . . . . 5 class 𝒫 𝑥
97, 2, 8wral 3055 . . . 4 wff 𝑦 ∈ 𝒫 𝑥 𝑦𝑥
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1532 . . . . . . . 8 class 𝑧
123, 11cin 3942 . . . . . . 7 class (𝑦𝑧)
1312, 6wcel 2098 . . . . . 6 wff (𝑦𝑧) ∈ 𝑥
1413, 10, 6wral 3055 . . . . 5 wff 𝑧𝑥 (𝑦𝑧) ∈ 𝑥
1514, 2, 6wral 3055 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥
169, 15wa 395 . . 3 wff (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)
1716, 5cab 2703 . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
181, 17wceq 1533 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  istopg  22748
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