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Definition df-t0 17258
Description: Define T0 or Kolmogorov spaces. A T0 space satisfies a kind of "topological extensionality" principle (compare ax-ext 2347): 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 17292) 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 17251 . 2  class  Kol2
2 vx . . . . . . . . 9  set  x
3 vo . . . . . . . . 9  set  o
42, 3wel 1716 . . . . . . . 8  wff  x  e.  o
5 vy . . . . . . . . 9  set  y
65, 3wel 1716 . . . . . . . 8  wff  y  e.  o
74, 6wb 176 . . . . . . 7  wff  ( x  e.  o  <->  y  e.  o )
8 vj . . . . . . . 8  set  j
98cv 1646 . . . . . . 7  class  j
107, 3, 9wral 2628 . . . . . 6  wff  A. o  e.  j  ( x  e.  o  <->  y  e.  o )
112, 5weq 1648 . . . . . 6  wff  x  =  y
1210, 11wi 4 . . . . 5  wff  ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
139cuni 3929 . . . . 5  class  U. j
1412, 5, 13wral 2628 . . . 4  wff  A. y  e.  U. j ( A. o  e.  j  (
x  e.  o  <->  y  e.  o )  ->  x  =  y )
1514, 2, 13wral 2628 . . 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 16848 . . 3  class  Top
1715, 8, 16crab 2632 . 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 1647 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  17265
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