2. Let f and g be continous mappings on a metric space X into a metric space Y and E be a dense subsetof X(a) Prove that f (E) is dense in f (X)(b) If f (p)-g (p) for all p E, then f (p)g (p) for all p eX3. Let (an) be a sequence in a metric space X.(a) Prove if X is compact, then xn → x if and only if every convergent subsequence of {z») convergesto 1.(b) Find a counterexample showing that (a) may not hold when X is not compact.

Note: As per rules, the three questions 2(a), 2(b) and 3(a) are solved. The problems concern…

Answered: 2. Let f and g be continous mappings on…

Elements Of Modern Algebra

8th Edition

ISBN:9781285463230

Author:Gilbert, Linda, Jimmie

Publisher:Gilbert, Linda, Jimmie

Chapter1: Fundamentals

Section1.7: Relations

Problem 23E

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