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Definition df-t0 17331
Description: Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2385): 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 17365) 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 17324 . 2  class  Kol2
2 vx . . . . . . . . 9  set  x
3 vo . . . . . . . . 9  set  o
42, 3wel 1722 . . . . . . . 8  wff  x  e.  o
5 vy . . . . . . . . 9  set  y
65, 3wel 1722 . . . . . . . 8  wff  y  e.  o
74, 6wb 177 . . . . . . 7  wff  ( x  e.  o  <->  y  e.  o )
8 vj . . . . . . . 8  set  j
98cv 1648 . . . . . . 7  class  j
107, 3, 9wral 2666 . . . . . 6  wff  A. o  e.  j  ( x  e.  o  <->  y  e.  o )
112, 5weq 1650 . . . . . 6  wff  x  =  y
1210, 11wi 4 . . . . 5  wff  ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
139cuni 3975 . . . . 5  class  U. j
1412, 5, 13wral 2666 . . . 4  wff  A. y  e.  U. j ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
1514, 2, 13wral 2666 . . 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 16913 . . 3  class  Top
1715, 8, 16crab 2670 . 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 1649 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 set class
This definition is referenced by:  ist0  17338
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