Which of the following metric spaces are complete?
$X_1=(0,\frac{\pi}{2}), d(x,y)=|\tan x-\tan y|$
$X_2=[0,1], d(x,y)=\frac{|x-y|}{1+|x-y|}$
$X_3=\mathbb{Q}, d(x,y)=1\forall x\neq y$
$X_4=\mathbb{R}, d(x,y)=|e^x-e^y|$
My answer : option $2$ and $3 $ only .
Here is the link of question see $3.9 $ https://www.imsc.res.in/~office/nbhm/qp/nbhmra06.pdf
answer key link : see 3. 9 https://www.imsc.res.in/~office/nbhm/qp/nbhmra06key.pdf
My confusion is that they have given option $1,2,3 $
I Just need conformation that answer key right or wrong