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Definition df-top 20463
Description: Define the (proper) class of all topologies. See istop2g 20468 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 20459 . 2 class Top
2 vy . . . . . . . 8 setvar 𝑦
32cv 1473 . . . . . . 7 class 𝑦
43cuni 4366 . . . . . 6 class 𝑦
5 vx . . . . . . 7 setvar 𝑥
65cv 1473 . . . . . 6 class 𝑥
74, 6wcel 1976 . . . . 5 wff 𝑦𝑥
86cpw 4107 . . . . 5 class 𝒫 𝑥
97, 2, 8wral 2895 . . . 4 wff 𝑦 ∈ 𝒫 𝑥 𝑦𝑥
10 vz . . . . . . . . 9 setvar 𝑧
1110cv 1473 . . . . . . . 8 class 𝑧
123, 11cin 3538 . . . . . . 7 class (𝑦𝑧)
1312, 6wcel 1976 . . . . . 6 wff (𝑦𝑧) ∈ 𝑥
1413, 10, 6wral 2895 . . . . 5 wff 𝑧𝑥 (𝑦𝑧) ∈ 𝑥
1514, 2, 6wral 2895 . . . 4 wff 𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥
169, 15wa 382 . . 3 wff (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)
1716, 5cab 2595 . 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
181, 17wceq 1474 1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥 𝑦𝑥 ∧ ∀𝑦𝑥𝑧𝑥 (𝑦𝑧) ∈ 𝑥)}
Colors of variables: wff setvar class
This definition is referenced by:  istopg  20467
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