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Definition df-t0 17377
Description: Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2417): 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 17411) 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 17370 . 2  class  Kol2
2 vx . . . . . . . . 9  set  x
3 vo . . . . . . . . 9  set  o
42, 3wel 1726 . . . . . . . 8  wff  x  e.  o
5 vy . . . . . . . . 9  set  y
65, 3wel 1726 . . . . . . . 8  wff  y  e.  o
74, 6wb 177 . . . . . . 7  wff  ( x  e.  o  <->  y  e.  o )
8 vj . . . . . . . 8  set  j
98cv 1651 . . . . . . 7  class  j
107, 3, 9wral 2705 . . . . . 6  wff  A. o  e.  j  ( x  e.  o  <->  y  e.  o )
112, 5weq 1653 . . . . . 6  wff  x  =  y
1210, 11wi 4 . . . . 5  wff  ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
139cuni 4015 . . . . 5  class  U. j
1412, 5, 13wral 2705 . . . 4  wff  A. y  e.  U. j ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
1514, 2, 13wral 2705 . . 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 16958 . . 3  class  Top
1715, 8, 16crab 2709 . 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 1652 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  17384
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