Definition df-top 12537

Description: Define the class of topologies. It is a proper class. See istopg 12538 and istopfin 12539 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 12536	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1341	. . . . . . 7 class y
4	3	cuni 3783	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1341	. . . . . 6 class x
7	4, 6	wcel 2135	. . . . 5 wff U. y e. x
8	6	cpw 3553	. . . . 5 class ~P x
9	7, 2, 8	wral 2442	. . . 4 wff A. y e. ~P x U. y e. x
10		vz	. . . . . . . . 9 setvar z
11	10	cv 1341	. . . . . . . 8 class z
12	3, 11	cin 3110	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 2135	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2442	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2442	. . . 4 wff A. y e. x A. z e. x ( y i^i z ) e. x
16	9, 15	wa 103	. . 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 2150	. 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 1342	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  12538