Tutorial T0 Space


T0 Space


   


A topological space X is said to be a T0-space if for any pair of distinct points of X, there exist at least one open set which contains one of them but not the other.
In other words, a topological space X is said to be a T0-space if for any , there exist an open set U such that x belongs to U but y does not belongs to U.

Example:
Let X={a,, b, c} with topology defined on X, then (X,T) is a T0-space, because

  • for a and b, there exist an open set such that and
  • for a and c, there exist an open set such that and
  • for b and c, there exist an open set such that and

Theorems:

  • Let X is an indiscrete topological space with at least two points, then X is not a T0-space.
  • Let X is a discrete topological space with at least two points, then X is not a T0-space.
  • The real line R with usual topology is T0-space.
  • Every sub space of T0-space is T0-space.
  • A topological space X is T0-space if and only if, for any .
  • Every two point co-finite topological space is a T0-space.
  • Every two point co-countable topological space is a T0-space.
  • If each singleton subset of a two point topological space is closed, then it is T0-space.
  • If each finite subset of a two point topological space is closed, then it is a T0-space.
  • Any homeomorphic image of a T0-space is a T0-space.
  • A pseudo metric space is a metric space if and only if it is a T0-space.



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