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Definition df-top 22915
Description: Define the class of topologies. It is a proper class (see topnex 23018). See istopg 22916 and istop2g 22917 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 22914 . 2 class Top
2 vy . . . . . . . 8 setvar 𝑦
32cv 1535 . . . . . . 7 class 𝑦
43cuni 4911 . . . . . 6 class 𝑦
5 vx . . . . . . 7 setvar 𝑥
65cv 1535 . . . . . 6 class 𝑥
74, 6wcel 2105 . . . . 5 wff 𝑦𝑥
86cpw 4604 . . . . 5 class 𝒫 𝑥
97, 2, 8wral 3058 . . . 4 wff 𝑦 ∈ 𝒫 𝑥 𝑦𝑥
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1535 . . . . . . . 8 class 𝑧
123, 11cin 3961 . . . . . . 7 class (𝑦𝑧)
1312, 6wcel 2105 . . . . . 6 wff (𝑦𝑧) ∈ 𝑥
1413, 10, 6wral 3058 . . . . 5 wff 𝑧𝑥 (𝑦𝑧) ∈ 𝑥
1514, 2, 6wral 3058 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥
169, 15wa 395 . . 3 wff (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)
1716, 5cab 2711 . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
181, 17wceq 1536 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  istopg  22916
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