Page 177 - DMTH503_TOPOLOGY

Unit 18: Normal Spaces, Regular Spaces and Completely Regular Spaces

Solution: Let (X, T) be a topological space for which the given conditions hold. Let F be a T-closed Notes
subset of X and let x be a point of X such that xF. Then X – F is a T-open set containing x. By the
given condition there exits a continuous mapping f : X[0, 1] such that
f(x) = 0 and f(Y) = 1  y  X – (X – F) i.e. y  F.
Hence the space is completely regular.
Conversely, Let (X, T) be a completely regular space and let G be an open subset of X containing x.

Then X – G is a closed subset of X such that x  X – G. Since X is completely regular there exists
a continuous mapping f : X  [0, 1] such that
f(x) = 0 and f(X – G) = {1}

Self Assessment

8. Let F be a closed subset of a completely regular space (X, T) and x F, then prove that
0
there exists a continuous map f : X[0, 1] s.t. f(x ) = 1, f(F) = {0}.
0
9. Prove that a normal space is completely regular iff it is regular.
18.4 Summary

 A topological space (X, T) is said to be normal space if: given a pair of disjoint closed sets
C , C X.disjoint open sets G , G X s.t. C G , C G .
1 2 1 2 1 1 2 2
 Matric spaces are normal.

 A closed subspace of a normal space is a normal space.
 A topological space (X, T) is said to be regular space if: given an element xX and closed
set FX s.t. xF,disjoint open sets G , G X s.t. xG , FG .
1 2 1 2
 A regular T -space is called a T -space.
1 3
 A normal T -space is called a T -space.
1 4
 A normal space is a regular.
 A topological space (X, T) is called a completely regular space if : given a closed set FX
and a point xX s.t. xF,a continuous map f : X[0, 1] with the property, f(x) = 0,
f(F) = {1}.
 Every metric space is a completely regular space.
 Complete regularity is hereditary property.

 A completely regular space is regular.
18.5 Keywords

Compact: A topological space (X, T) is called compact if every open cover of X has a finite sub
cover.
Hausdorff Space: It is a topological space in which each pair of distinct points can be separated
by disjoint neighbourhoods.
Metric Space: Any metric space is a topological space, the topology being the set of all open sets.
Tychonoff Space: Tychonoff space is a Hausdorff space (X, T) in which any closed set A and any
xA are functionally separated.

LOVELY PROFESSIONAL UNIVERSITY 171