Page 96 - DMTH503_TOPOLOGY

Topology

Notes which is open, being a union of two open sets.
Hence [x, y] is closed.

Example 5: Give an example of two closed subsets A and B of the real line  such that
d(A, B) = 0 but A  B = .
Solution: Let A = {2, 3, 4, 5, …}

1
1
1
B = 2 , 3 , 4 ,  ,
2 3 4
Clearly A  B = .
d(A, B) = inf{d (x, y) : x  A, y  B}
If n  A and n  1  B
n
d(A, B) = lim d n, n  1 n 
n
= lim n 1 [ d is usual metric for ]
n
= 0

9.1.7 Interior, Closure and Boundary of a Point

Interior

Let (X, d) be a metric space and A X.
A point x A is called an interior point of A if r R s.t. S A.
+
r (x)
The set of all interior point of A is called the interior of A and is denoted by A°, or by Int. (A).
Thus A° = int. (A) = {x A : S A for some r)
r (x)
Alternatively, we define
A° = (S : S A).
r (x) r (x)
Evidently
(i) A° is an open set.

For an arbitrary union of open sets is open.
(ii) A° is the largest open subset of A.

Closure

Let (X, d) be a metric space and A X.
The closure of A, denoted by A , is defined as the intersection of all closed sets that contain A.
Symbolically
A = {F X : F is closed, F A} ...(1)
Evidently
(i) A is closed set

For an arbitrary intersection of closed sets is closed.

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