Definition df-top 14855

Description: Define the class of topologies. It is a proper class. See istopg 14856 and istopfin 14857 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 14854	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1397	. . . . . . 7 class y
4	3	cuni 3913	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1397	. . . . . 6 class x
7	4, 6	wcel 2203	. . . . 5 wff U. y e. x
8	6	cpw 3668	. . . . 5 class ~P x
9	7, 2, 8	wral 2520	. . . 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 3209	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 2203	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2520	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2520	. . . 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 2218	. 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  14856