Definition df-top 23212

Description: Define the class of topologies. It is a proper class (see topnex 23314). See istopg 23213 and istop2g 23214 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 23211	. 2 class Top
2		vy	. . . . . . . 8 setvar 𝑦
3	2	cv 1568	. . . . . . 7 class 𝑦
4	3	cuni 4866	. . . . . 6 class ∪ 𝑦
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1568	. . . . . 6 class 𝑥
7	4, 6	wcel 2144	. . . . 5 wff ∪ 𝑦 ∈ 𝑥
8	6	cpw 4556	. . . . 5 class 𝒫 𝑥
9	7, 2, 8	wral 3076	. . . 4 wff ∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥
10		vz	. . . . . . . . 9 setvar 𝑧
11	10	cv 1568	. . . . . . . 8 class 𝑧
12	3, 11	cin 4089	. . . . . . 7 class (𝑦 ∩ 𝑧)
13	12, 6	wcel 2144	. . . . . 6 wff (𝑦 ∩ 𝑧) ∈ 𝑥
14	13, 10, 6	wral 3076	. . . . 5 wff ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥
15	14, 2, 6	wral 3076	. . . 4 wff ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥
16	9, 15	wa 401	. . 3 wff (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)
17	16, 5	cab 2738	. 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
18	1, 17	wceq 1569	1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}

This definition is used by:  istopg  23213