Page 98 - DMTH503_TOPOLOGY

Topology

Notes 1 1 1 1
 
(ii) If A   1, , , ,....  : n N 

2 3 4 n
then 0 is the limit point of A
1
For lim  0
n n

9.1.8 Neighborhood

Let (X, d) be a metric space and x X, A X
A subset A of X is called a neighborhood (nbd) of x if  open sphere S s.t. S A
r (x) r (x)
This means that A is nbd of a point x iff x is an interior point of A.
From the definition of nbd, it is clear that:

(1) Every superset of a nbd of a point is also a nbd.
(2) Every open sphere S is a nbd of x.
r (x)
(3) Every closed sphere S is a nbd of x.
r (x)
(4) Intersection of two nbds of the same point is given a nbd of that point.

(5) A set is open if it contains a nbd of each of its points.
(6) Nbd of a point need not be an open set.

9.1.9 Theorems and Solved Examples

Theorem 5: A subset of a metric space is open iff it is a nbd of each of its point.
Proof: Let A be a subset of a metric space (X, d).

Step I: Given A is a nbd of each of its points.
Aim: A is an open set
Recall that a set N is called nbd of a point x X if open set G X s.t. x  G N.
Let p A be arbitrary, then by assumption, A is a nbd of p. By definition of nbd, open set
G  X s.t. p G A.
p p
It is true p A 

Take A = {G : p G , G is an open set, G A }
p p p p
= An arbitrary union of open sets

= open set
A is an open set.
Step II: Let A be an open subset of X.
Aim: A is a nbd of each of its points. By assumption, we can write p A A  p A .

A is a nbd of each of its points.
Problem: Every set of discrete metric space is open.

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