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Definition df-top 20521
 Description: Define the (proper) class of all topologies. See istop2g 20526 for an alternate way to express finite intersection. 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 20517 . 2 class Top
2 vy . . . . . . . 8 setvar 𝑦
32cv 1474 . . . . . . 7 class 𝑦
43cuni 4372 . . . . . 6 class 𝑦
5 vx . . . . . . 7 setvar 𝑥
65cv 1474 . . . . . 6 class 𝑥
74, 6wcel 1977 . . . . 5 wff 𝑦𝑥
86cpw 4108 . . . . 5 class 𝒫 𝑥
97, 2, 8wral 2896 . . . 4 wff 𝑦 ∈ 𝒫 𝑥 𝑦𝑥
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1474 . . . . . . . 8 class 𝑧
123, 11cin 3539 . . . . . . 7 class (𝑦𝑧)
1312, 6wcel 1977 . . . . . 6 wff (𝑦𝑧) ∈ 𝑥
1413, 10, 6wral 2896 . . . . 5 wff 𝑧𝑥 (𝑦𝑧) ∈ 𝑥
1514, 2, 6wral 2896 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥
169, 15wa 383 . . 3 wff (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)
1716, 5cab 2596 . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
181, 17wceq 1475 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
 Colors of variables: wff setvar class This definition is referenced by:  istopg  20525
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