Definition df-top 12636

Description: Define the class of topologies. It is a proper class. See istopg 12637 and istopfin 12638 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 12635	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1342	. . . . . . 7 class y
4	3	cuni 3789	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1342	. . . . . 6 class x
7	4, 6	wcel 2136	. . . . 5 wff U. y e. x
8	6	cpw 3559	. . . . 5 class ~P x
9	7, 2, 8	wral 2444	. . . 4 wff A. y e. ~P x U. y e. x
10		vz	. . . . . . . . 9 setvar z
11	10	cv 1342	. . . . . . . 8 class z
12	3, 11	cin 3115	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 2136	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2444	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2444	. . . 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 2151	. 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 1343	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  12637