Page 97 - DMTH503_TOPOLOGY

Unit 9: The Metric Topology

(ii) A A. Notes
(iii) A is the smallest closed set which contain A.
Alternatively we define

A = A D (A) ...(2)
A point x  A is called a point of closure of A.
Alternatively, a point x X is called a point of closure of A iff x A or x D (A).

Boundary of a Point

Let (X, d) be a metric space. Let A X

(i) Boundary or Frontier of a set A is denoted by b(A) or F (A) and is defined as
r
b(A) = F (A) = X – A° (X – A)°.
r
Elements of b(A) are called boundary points of A.
(ii) The exterior of A is defied as the set (X – A)° and is denoted by ext (A).

Symbolically ext (A) = (X – A)°.
(iii) A is said to be dense or everywhere dense in X if A  X.

(iv) A is said to be somewhere dense if (A)   i.e., if closure of A contains some open set.
(v) A is said to be nowhere dense (or non where dense set) if (A)   .

(vi) A metric space (X, d) is said to be separable if A X s.t. A is countable and A  X 
(vii) A is said to be dense in itself of A D (A).

Example 6:
(1) To find the boundary of set of integers Z and set of rationals Q.
Z° =  {G R : G is open ad G Z} = 

For every sub set of R contains fractions also.
Similarly (R – Z)° = 
b (Z) = R – Z° (R – Z)° = R – = R
b (Z) = R.

Similarly b(Q) = R.
(2) Give two examples of limit points
 1 

(i) If A  1 n : n N ,

 3 4 5 6 
i.e. A  2, , , , ,.... , then
2 3 4 5
 1 
1 is limit point of A. For lim 1   1.

n  n 

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