[NBHM_2006_PhD Screening test_Topology]
Which of the following metric spaces are complete?
$X_1=(0,1), d(x,y)=|\tan x-\tan y|$
$X_2=[0,1], d(x,y)=\frac{|x-y|}{1+|x-y|}$
$X_3=\mathbb{Q}, d(x,y)=1\forall x\neq y$
$X_4=\mathbb{R}, d(x,y)=|e^x-e^y|$
$2$ is complete as closed subset of a complete metric space is complete and the metric is also equivalent to our usual metric.
$3$ is also complete as every Cauchy sequence is constant ultimately hence convergent.
$4$ is not complete I am sure but not able to find out a counter example, not sure about 1.thank you for help.
Best Answer
For (1), consider the sequence $\left\langle\frac1{2^n}:n\in\Bbb N\right\rangle$. Is it $d$-Cauchy? Does it converge to anything in $X_1$?
For (4), what about $\langle -n:n\in\Bbb N\rangle$?