Definition df-top 14585

Description: Define the class of topologies. It is a proper class. See istopg 14586 and istopfin 14587 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 14584	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1372	. . . . . . 7 class y
4	3	cuni 3864	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1372	. . . . . 6 class x
7	4, 6	wcel 2178	. . . . 5 wff U. y e. x
8	6	cpw 3626	. . . . 5 class ~P x
9	7, 2, 8	wral 2486	. . . 4 wff A. y e. ~P x U. y e. x
10		vz	. . . . . . . . 9 setvar z
11	10	cv 1372	. . . . . . . 8 class z
12	3, 11	cin 3173	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 2178	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2486	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2486	. . . 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 2193	. 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 1373	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  14586