Definition df-top 14470

Description: Define the class of topologies. It is a proper class. See istopg 14471 and istopfin 14472 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 14469	. 2 class Top
2		vy	. . . . . . . 8 setvar 𝑦
3	2	cv 1372	. . . . . . 7 class 𝑦
4	3	cuni 3850	. . . . . 6 class ∪ 𝑦
5		vx	. . . . . . 7 setvar 𝑥
6	5	cv 1372	. . . . . 6 class 𝑥
7	4, 6	wcel 2176	. . . . 5 wff ∪ 𝑦 ∈ 𝑥
8	6	cpw 3616	. . . . 5 class 𝒫 𝑥
9	7, 2, 8	wral 2484	. . . 4 wff ∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥
10		vz	. . . . . . . . 9 setvar 𝑧
11	10	cv 1372	. . . . . . . 8 class 𝑧
12	3, 11	cin 3165	. . . . . . 7 class (𝑦 ∩ 𝑧)
13	12, 6	wcel 2176	. . . . . 6 wff (𝑦 ∩ 𝑧) ∈ 𝑥
14	13, 10, 6	wral 2484	. . . . 5 wff ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥
15	14, 2, 6	wral 2484	. . . 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 2191	. 2 class {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}
18	1, 17	wceq 1373	1 wff Top = {𝑥 ∣ (∀𝑦 ∈ 𝒫 𝑥∪ 𝑦 ∈ 𝑥 ∧ ∀𝑦 ∈ 𝑥 ∀𝑧 ∈ 𝑥 (𝑦 ∩ 𝑧) ∈ 𝑥)}

This definition is referenced by:  istopg  14471