Definition df-top 15022

Description: Define the class of topologies. It is a proper class. See istopg 15023 and istopfin 15024 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 15021	. 2 class Top
2		vy	. . . . . . . 8 setvar y
3	2	cv 1401	. . . . . . 7 class y
4	3	cuni 3930	. . . . . 6 class U. y
5		vx	. . . . . . 7 setvar x
6	5	cv 1401	. . . . . 6 class x
7	4, 6	wcel 2209	. . . . 5 wff U. y e. x
8	6	cpw 3685	. . . . 5 class ~P x
9	7, 2, 8	wral 2528	. . . 4 wff A. y e. ~P x U. y e. x
10		vz	. . . . . . . . 9 setvar z
11	10	cv 1401	. . . . . . . 8 class z
12	3, 11	cin 3219	. . . . . . 7 class ( y i^i z )
13	12, 6	wcel 2209	. . . . . 6 wff ( y i^i z ) e. x
14	13, 10, 6	wral 2528	. . . . 5 wff A. z e. x ( y i^i z ) e. x
15	14, 2, 6	wral 2528	. . . 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 2224	. 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 1402	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  15023