Download Definition : a topological space (X,T) is said to be completely regular

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}
Definition : a tychonoff space or
space is completely
regular T1-space
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
( )∩
( ) = ∅,
→
ℎ ( )=0∈
( )
∈
( ) = {1} ⊂
∋⇒
( )
⊂
( ),
( )
Thus there exist disjoint T-open sets
containing x and F 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
ℎ
ℎ :
→ [0,1]
ℎ ℎ ( ) = 0
( ) = 1
ℎ
ℎ
∈ ∈
)
→ [0,1]
ℎ ℎ ({ }) = {0}
:
( ) = 1.
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 .