Definition df-top 22855

Description: Define the class of topologies. It is a proper class (see topnex 22957). See istopg 22856 and istop2g 22857 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 22854	. 2 class Top
2		vy	. . . . . . . 8 setvar 𝑦
3	2	cv 1541	. . . . . . 7 class 𝑦
4	3	cuni 4865	. . . . . 6 class ∪ 𝑦
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1541	. . . . . 6 class 𝑥
7	4, 6	wcel 2114	. . . . 5 wff ∪ 𝑦 ∈ 𝑥
8	6	cpw 4556	. . . . 5 class 𝒫 𝑥
9	7, 2, 8	wral 3052	. . . 4 wff ∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥
10		vz	. . . . . . . . 9 setvar 𝑧
11	10	cv 1541	. . . . . . . 8 class 𝑧
12	3, 11	cin 3902	. . . . . . 7 class (𝑦 ∩ 𝑧)
13	12, 6	wcel 2114	. . . . . 6 wff (𝑦 ∩ 𝑧) ∈ 𝑥
14	13, 10, 6	wral 3052	. . . . 5 wff ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥
15	14, 2, 6	wral 3052	. . . 4 wff ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥
16	9, 15	wa 395	. . 3 wff (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)
17	16, 5	cab 2715	. 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
18	1, 17	wceq 1542	1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}

This definition is referenced by:  istopg  22856