2014-54.md

course	course_year	question_number	tags	title	year
Metric and Topological Spaces	IB	54	IB 2014 Metric and Topological Spaces	Paper 3, Section I, E	2014

Suppose $(X, d)$ is a metric space. Do the following necessarily define a metric on $X$ ? Give proofs or counterexamples.

(i) $d_{1}(x, y)=k d(x, y)$ for some constant $k>0$, for all $x, y \in X$.

(ii) $d_{2}(x, y)=\min {1, d(x, y)}$ for all $x, y \in X$.

(iii) $d_{3}(x, y)=(d(x, y))^{2}$ for all $x, y \in X$.