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Definition df-top 20908
Description: Define the class of topologies. It is a proper class (see topnex 21010). See istopg 20909 and istop2g 20910 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 20907 . 2 class Top
2 vy . . . . . . . 8 setvar 𝑦
32cv 1636 . . . . . . 7 class 𝑦
43cuni 4630 . . . . . 6 class 𝑦
5 vx . . . . . . 7 setvar 𝑥
65cv 1636 . . . . . 6 class 𝑥
74, 6wcel 2156 . . . . 5 wff 𝑦𝑥
86cpw 4351 . . . . 5 class 𝒫 𝑥
97, 2, 8wral 3096 . . . 4 wff 𝑦 ∈ 𝒫 𝑥 𝑦𝑥
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1636 . . . . . . . 8 class 𝑧
123, 11cin 3768 . . . . . . 7 class (𝑦𝑧)
1312, 6wcel 2156 . . . . . 6 wff (𝑦𝑧) ∈ 𝑥
1413, 10, 6wral 3096 . . . . 5 wff 𝑧𝑥 (𝑦𝑧) ∈ 𝑥
1514, 2, 6wral 3096 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥
169, 15wa 384 . . 3 wff (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)
1716, 5cab 2792 . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
181, 17wceq 1637 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  istopg  20909
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