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Definition df-t0 18892
Description: Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2419): any two points which are members of the same open sets are equal, or in contraposition, for any two distinct points there is an open set which contains one point but not the other. This differs from T1 spaces (see ist1-2 18926) in that in a T1 space you can choose which point will be in the open set and which outside; in a T0 space you only know that one of the two points is in the set. (Contributed by Jeff Hankins, 1-Feb-2010.)
Assertion
Ref Expression
df-t0  |-  Kol2  =  { j  e.  Top  | 
A. x  e.  U. j A. y  e.  U. j ( A. o  e.  j  ( x  e.  o  <->  y  e.  o )  ->  x  =  y ) }
Distinct variable group:    j, o, x, y

Detailed syntax breakdown of Definition df-t0
StepHypRef Expression
1 ct0 18885 . 2  class  Kol2
2 vx . . . . . . . . 9  setvar  x
3 vo . . . . . . . . 9  setvar  o
42, 3wel 1757 . . . . . . . 8  wff  x  e.  o
5 vy . . . . . . . . 9  setvar  y
65, 3wel 1757 . . . . . . . 8  wff  y  e.  o
74, 6wb 184 . . . . . . 7  wff  ( x  e.  o  <->  y  e.  o )
8 vj . . . . . . . 8  setvar  j
98cv 1368 . . . . . . 7  class  j
107, 3, 9wral 2710 . . . . . 6  wff  A. o  e.  j  ( x  e.  o  <->  y  e.  o )
112, 5weq 1695 . . . . . 6  wff  x  =  y
1210, 11wi 4 . . . . 5  wff  ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
139cuni 4086 . . . . 5  class  U. j
1412, 5, 13wral 2710 . . . 4  wff  A. y  e.  U. j ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
1514, 2, 13wral 2710 . . 3  wff  A. x  e.  U. j A. y  e.  U. j ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
16 ctop 18473 . . 3  class  Top
1715, 8, 16crab 2714 . 2  class  { j  e.  Top  |  A. x  e.  U. j A. y  e.  U. j
( A. o  e.  j  ( x  e.  o  <->  y  e.  o )  ->  x  =  y ) }
181, 17wceq 1369 1  wff  Kol2  =  { j  e.  Top  | 
A. x  e.  U. j A. y  e.  U. j ( A. o  e.  j  ( x  e.  o  <->  y  e.  o )  ->  x  =  y ) }
Colors of variables: wff setvar class
This definition is referenced by:  ist0  18899
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