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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 .