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Definition df-top 12755
Description: Define the class of topologies. It is a proper class. See istopg 12756 and istopfin 12757 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 12754 . 2 class Top
2 vy . . . . . . . 8 setvar 𝑦
32cv 1347 . . . . . . 7 class 𝑦
43cuni 3794 . . . . . 6 class 𝑦
5 vx . . . . . . 7 setvar 𝑥
65cv 1347 . . . . . 6 class 𝑥
74, 6wcel 2141 . . . . 5 wff 𝑦𝑥
86cpw 3564 . . . . 5 class 𝒫 𝑥
97, 2, 8wral 2448 . . . 4 wff 𝑦 ∈ 𝒫 𝑥 𝑦𝑥
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1347 . . . . . . . 8 class 𝑧
123, 11cin 3120 . . . . . . 7 class (𝑦𝑧)
1312, 6wcel 2141 . . . . . 6 wff (𝑦𝑧) ∈ 𝑥
1413, 10, 6wral 2448 . . . . 5 wff 𝑧𝑥 (𝑦𝑧) ∈ 𝑥
1514, 2, 6wral 2448 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥
169, 15wa 103 . . 3 wff (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)
1716, 5cab 2156 . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
181, 17wceq 1348 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
Colors of variables: wff set class
This definition is referenced by:  istopg  12756
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