Definition df-top 13794

Description: Define the class of topologies. It is a proper class. See istopg 13795 and istopfin 13796 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 13793	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1362	. . . . . . 7 class y
4	3	cuni 3821	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1362	. . . . . 6 class x
7	4, 6	wcel 2158	. . . . 5 wff U. y e. x
8	6	cpw 3587	. . . . 5 class ~P x
9	7, 2, 8	wral 2465	. . . 4 wff A. y e. ~P x U. y e. x
10		vz	. . . . . . . . 9 setvar z
11	10	cv 1362	. . . . . . . 8 class z
12	3, 11	cin 3140	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 2158	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2465	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2465	. . . 4 wff A. y e. x A. z e. x ( y i^i z ) e. x
16	9, 15	wa 104	. . 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 2173	. 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 1363	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  13795