Definition df-top 13537

Description: Define the class of topologies. It is a proper class. See istopg 13538 and istopfin 13539 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 13536	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1352	. . . . . . 7 class y
4	3	cuni 3811	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1352	. . . . . 6 class x
7	4, 6	wcel 2148	. . . . 5 wff U. y e. x
8	6	cpw 3577	. . . . 5 class ~P x
9	7, 2, 8	wral 2455	. . . 4 wff A. y e. ~P x U. y e. x
10		vz	. . . . . . . . 9 setvar z
11	10	cv 1352	. . . . . . . 8 class z
12	3, 11	cin 3130	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 2148	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2455	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2455	. . . 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 2163	. 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 1353	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  13538