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Definition df-t0 17041
Description: Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2264): 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 17075) 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 17034 . 2  class  Kol2
2 vx . . . . . . . . 9  set  x
3 vo . . . . . . . . 9  set  o
42, 3wel 1685 . . . . . . . 8  wff  x  e.  o
5 vy . . . . . . . . 9  set  y
65, 3wel 1685 . . . . . . . 8  wff  y  e.  o
74, 6wb 176 . . . . . . 7  wff  ( x  e.  o  <->  y  e.  o )
8 vj . . . . . . . 8  set  j
98cv 1622 . . . . . . 7  class  j
107, 3, 9wral 2543 . . . . . 6  wff  A. o  e.  j  ( x  e.  o  <->  y  e.  o )
112, 5weq 1624 . . . . . 6  wff  x  =  y
1210, 11wi 4 . . . . 5  wff  ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
139cuni 3827 . . . . 5  class  U. j
1412, 5, 13wral 2543 . . . 4  wff  A. y  e.  U. j ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
1514, 2, 13wral 2543 . . 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 16631 . . 3  class  Top
1715, 8, 16crab 2547 . 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 1623 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  17048
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