Definition df-top 14234

Description: Define the class of topologies. It is a proper class. See istopg 14235 and istopfin 14236 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∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}

Distinct variable group:   𝑥,𝑦,𝑧

Detailed syntax breakdown of Definition df-top

Step	Hyp	Ref	Expression
1		ctop 14233	. 2 class Top
2		vy	. . . . . . . 8 setvar 𝑦
3	2	cv 1363	. . . . . . 7 class 𝑦
4	3	cuni 3839	. . . . . 6 class ∪ 𝑦
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1363	. . . . . 6 class 𝑥
7	4, 6	wcel 2167	. . . . 5 wff ∪ 𝑦 ∈ 𝑥
8	6	cpw 3605	. . . . 5 class 𝒫 𝑥
9	7, 2, 8	wral 2475	. . . 4 wff ∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥
10		vz	. . . . . . . . 9 setvar 𝑧
11	10	cv 1363	. . . . . . . 8 class 𝑧
12	3, 11	cin 3156	. . . . . . 7 class (𝑦 ∩ 𝑧)
13	12, 6	wcel 2167	. . . . . 6 wff (𝑦 ∩ 𝑧) ∈ 𝑥
14	13, 10, 6	wral 2475	. . . . 5 wff ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥
15	14, 2, 6	wral 2475	. . . 4 wff ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥
16	9, 15	wa 104	. . 3 wff (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)
17	16, 5	cab 2182	. 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
18	1, 17	wceq 1364	1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}

This definition is referenced by:  istopg  14235