Page 19 - DMTH503_TOPOLOGY

Unit 1: Topological Spaces

Notes
Example 15: If T = {, {a}, {a, b}, {a, c, d}, {a, b, e}, {a, b, c, d}, X} be a topology on
X = {a, b, c, d, e} then which of the set {a}, {b}, {c, e} are dense in X.

Solution: A is called dense in X if A = X (By definition)
{ a } =  {F : F is closed subset s.t. F  {a}} = X.

{ b } = X  {b, c, d, e}  {b, e} = {b, e}

{ c,e } = X  {b, c, d, e}  {c, d, e} = {c, d, e}
This shows that {a} is the only dense set in X.

Definition
 A is said to be dense in itself if A  D (A).
 A is said to be nowhere dense set in X if int ( A ) =  i.e., if the interior of the closure of A
is an empty set.

1.5.2 Boundary Set

The Boundary set of A is the set of all those points which belong neither to the interior of A nor
to the interior of its complement and is denoted by b(A).
Symbolically, b(A) = X – A°  (X – A)°.

Elements of b(A) are called bounding points of A. Boundary points are, sometimes called frontier
points.

Example 16: Define nowhere dense set and give an example of it.
Solution: D(N) = 

For if a is any real number, then consider a real number  > o, so small that open set (a – , a + )
does not contain any point of N.
 Z = {n : n  N}  {0}  {–n : n  N}

D(Z) = D {n : n  N}  D {0}  D {–n : n  N}
=     
=   Z
 D(Z)  Z  Z is closed.

 Z = Z
Int ( Z ) = Int (Z) =  {G  R : G is open, G  Z}
= 
 An open subset of R will be an open interval, say G = (a , a ). This open interval contains all
1 2
real numbers (rationals and irrationals) x s.t. a < x < a and therefore G  Z.
1 2
 Int (Z) = 
This proves that Z is nowhere dense set in R.

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