Page 92 - DMTH503_TOPOLOGY

Topology

Notes Thus for a pseudo metric
x = y d(x, y) = 0
but not conversely.
Remark: Thus every metric space is a pseudo metric space but every pseudo metric space is not
necessarily metric space.

Example 2: Consider a map d : RRR s.t.
2
2
d(x, y) = x - y  x, y  R
Evidently d(x, y) = 0  x = +y.
It can be shown that d is a pseudo metric but not metric.

9.1.3 Open and Closed Sphere

Let (X, ) be a metric space.
Let x  X and r  R . Then set { xX :(x , x) < r } is defined as open sphere (or simply sphere)
+
o o
with centre x and radius r.
o
The following have the same meaning:
Open sphere, closed sphere, open ball and open disc.
We denote this open sphere by the symbol S(x , r) or by S (x ) or by B (x , d) or B(x , r). This open
o r o r o o
sphere is also called as Spherical neighborhood of the point x or r-nhd of the point x .
o o
We denote closed sphere by S [x ] and is defined as
r o
S [x ] = { xX :(x, x )r }.
r o o
The following have the same meaning:
Closed sphere, closed ball, closed cell and disc.

Examples on Open Sphere

In case of usual metric, we see that
(i) If X = R, then S (x ) = (x – r, x + r) = open interval with x as centre.
r o o o o
(ii) If X = R , then S (x ) = open circle with centre x and radius r.
2
r o o
3
(iii) If X = R , then S (x ) = open sphere with centre x and radius r.
r o o
9.1.4 Boundary Set, Open Set, Limit Point and Closed Set
Boundary Set

Let (X, d) be a metric space and AX. A point x in X is called a boundary point of A if each open
sphere centered at x intersects A and A. The boundary of A is the set of all its boundary points
and is denoted by b(A). It has following properties.
(1) b(A) is a closed set
(2) b(A) = AA

(3) A is closedA contains its boundary.

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