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Definition df-top 12181
Description: Define the class of topologies. It is a proper class. See istopg 12182 and istopfin 12183 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 12180 . 2 class Top
2 vy . . . . . . . 8 setvar 𝑦
32cv 1330 . . . . . . 7 class 𝑦
43cuni 3736 . . . . . 6 class 𝑦
5 vx . . . . . . 7 setvar 𝑥
65cv 1330 . . . . . 6 class 𝑥
74, 6wcel 1480 . . . . 5 wff 𝑦𝑥
86cpw 3510 . . . . 5 class 𝒫 𝑥
97, 2, 8wral 2416 . . . 4 wff 𝑦 ∈ 𝒫 𝑥 𝑦𝑥
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1330 . . . . . . . 8 class 𝑧
123, 11cin 3070 . . . . . . 7 class (𝑦𝑧)
1312, 6wcel 1480 . . . . . 6 wff (𝑦𝑧) ∈ 𝑥
1413, 10, 6wral 2416 . . . . 5 wff 𝑧𝑥 (𝑦𝑧) ∈ 𝑥
1514, 2, 6wral 2416 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥
169, 15wa 103 . . 3 wff (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)
1716, 5cab 2125 . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
181, 17wceq 1331 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
Colors of variables: wff set class
This definition is referenced by:  istopg  12182
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