Definition df-top 14880

Description: Define the class of topologies. It is a proper class. See istopg 14881 and istopfin 14882 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 14879	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1397	. . . . . . 7 class y
4	3	cuni 3916	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1397	. . . . . 6 class x
7	4, 6	wcel 2205	. . . . 5 wff U. y e. x
8	6	cpw 3671	. . . . 5 class ~P x
9	7, 2, 8	wral 2522	. . . 4 wff A. y e. ~P x U. y e. x
10		vz	. . . . . . . . 9 setvar z
11	10	cv 1397	. . . . . . . 8 class z
12	3, 11	cin 3212	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 2205	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2522	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2522	. . . 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 2220	. 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 1398	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  14881