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١ Definition : a topological space (X,T) is said to be completely regular space iff every closed subset F of X and every point xخX-F there exist a continuous function f:X→[0,1] such that f(x)=0 , f(F)={1} space is completely regular T1-space Definition : a tychonoff space or Theorem : a topological space X is completely regular space iff ∀ ∈ ∀ ∈ ∋ ∈ ∃ : → [0,1] ∋ ( ) = 0 ( ) = 1, ∀ ∈ − Theorem: every completely regular space is regular space and then every tychonoff space is T3-space. Proof: let X is completely regular space .let F be aclosed subset of X and let x be appoint of X not in F that is xخX-F. By completely regular space , there exist a continuous map : → [0,1] such that f(x)=0 , f(F)={1}, also it is easy to see that the space [0,1] with the relative usual topology is a hausdorff space. Hence there exist open subset G and H of [0,1] such that 0 ,1 ∈ ∈ ∩ = ( ) ( ) ∅. , X such that ( )∩ ( ) = ∅, → ( ) = {1} ⊂ ∈ ℎ ( )=0∈ ( ) ∋⇒ ⊂ ( ) ٢ ( ), ( ) containing x and F Thus there exist disjoint T-open sets respectively it follows that X is regular .also since every tychonoff space is completely regular T1 space it follows that every tychonoff is T3 space. Theorem: every T4-space is tychonoff space : . . ℎ ℎ ( , ) ℎ ℎ ℎ ℎ − ℎ . . . . ℎ ℎ ∉ ℎ ℎ . { } ( , : , { } , ℎ ∈ → [0,1] ℎ ℎ ({ }) ) : ∈ = {0} ( ) = 1. ℎ ℎ ℎ ( ) = 0 ( ) = 1 ℎ ℎ , ℎ → [0,1] ٣ Theorem: completely regular space is topological property and then tychonoff space is topological property Theorem: completely regular space is hereditary property and then tychonoff is hereditary property .