Definition df-top 14672

Description: Define the class of topologies. It is a proper class. See istopg 14673 and istopfin 14674 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 14671	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1394	. . . . . . 7 class y
4	3	cuni 3888	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1394	. . . . . 6 class x
7	4, 6	wcel 2200	. . . . 5 wff U. y e. x
8	6	cpw 3649	. . . . 5 class ~P x
9	7, 2, 8	wral 2508	. . . 4 wff A. y e. ~P x U. y e. x
10		vz	. . . . . . . . 9 setvar z
11	10	cv 1394	. . . . . . . 8 class z
12	3, 11	cin 3196	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 2200	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2508	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2508	. . . 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 2215	. 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 1395	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  14673