Page 105 - DMTH503_TOPOLOGY

Unit 9: The Metric Topology

2. Let R be the set of all real numbers and let Notes

x y
d(x,y)  for all x, y R.


1 x y
Prove that d is a metric for R.
3. Every derived set in a metric space is a closed set.

4. Let A and B is disjoint closed set in a metric space (X, d). Then disjoint open sets G, H s.t.
A G, B H.
5. Let X and let d be a real function of ordered pairs of X which satisfies the following two
conditions:
d (x, y) = 0 x = y
and d(x, y) d (x, z) + d (z, y).
Show that d is a metric on X.

6. Give an example of a pseudo metric which is not metric.
7. Let X be a metric space. Show that every subset of X is open each subset of X which
consists of single point is open.

8. In a metric space prove that

(a) (A) Int(A ),

(b) A  {x: d(x,A) 0}.

9.5 Further Readings

Books B. Mendelson, Introduction to Topology, Dover Publication.
J. L. Kelly, General Topology, Van Nostrand, Reinhold Co., New York.

LOVELY PROFESSIONAL UNIVERSITY 99