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Definition df-top 11864
 Description: Define the class of topologies. It is a proper class. See istopg 11865 and istopfin 11866 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 11863 . 2 class Top
2 vy . . . . . . . 8 setvar 𝑦
32cv 1295 . . . . . . 7 class 𝑦
43cuni 3675 . . . . . 6 class 𝑦
5 vx . . . . . . 7 setvar 𝑥
65cv 1295 . . . . . 6 class 𝑥
74, 6wcel 1445 . . . . 5 wff 𝑦𝑥
86cpw 3449 . . . . 5 class 𝒫 𝑥
97, 2, 8wral 2370 . . . 4 wff 𝑦 ∈ 𝒫 𝑥 𝑦𝑥
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1295 . . . . . . . 8 class 𝑧
123, 11cin 3012 . . . . . . 7 class (𝑦𝑧)
1312, 6wcel 1445 . . . . . 6 wff (𝑦𝑧) ∈ 𝑥
1413, 10, 6wral 2370 . . . . 5 wff 𝑧𝑥 (𝑦𝑧) ∈ 𝑥
1514, 2, 6wral 2370 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥
169, 15wa 103 . . 3 wff (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)
1716, 5cab 2081 . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
181, 17wceq 1296 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
 Colors of variables: wff set class This definition is referenced by:  istopg  11865
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