Definition df-top 20699

Description: Define the class of topologies. It is a proper class (see topnex 20800). See istopg 20700 and istop2g 20701 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 | ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x ) }

Distinct variable group:   x, y, z

Detailed syntax breakdown of Definition df-top

Step	Hyp	Ref	Expression
1		ctop 20698	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1482	. . . . . . 7 class y
4	3	cuni 4436	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1482	. . . . . 6 class x
7	4, 6	wcel 1990	. . . . 5 wff U. y e. x
8	6	cpw 4158	. . . . 5 class ~P x
9	7, 2, 8	wral 2912	. . . 4 wff A. y e. ~P x U. y e. x
10		vz	. . . . . . . . 9 setvar z
11	10	cv 1482	. . . . . . . 8 class z
12	3, 11	cin 3573	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 1990	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2912	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2912	. . . 4 wff A. y e. x A. z e. x ( y i^i z ) e. x
16	9, 15	wa 384	. . 3 wff ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x )
17	16, 5	cab 2608	. 2 class { x | ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x ) }
18	1, 17	wceq 1483	1 wff Top = { x | ( A. y e. ~P x U. y e. x /\ A. y e. x A. z e. x ( y i^i z ) e. x ) }

This definition is referenced by:  istopg  20700