Define metric on $\mathbb R$ by $d_1(x,y)=\min(|x-y|,1)$. Give a example of a subsets that are closed and bounded, but not compact in the metric space with respect to this metric.\
I was thinking on using neighborhoods of radius 2 but I don't know if it works. Any suggestion or a example may be helpful.
[Math] Example of closed and bounded but not compact in R
analysis
Best Answer
The original question was:
But note that every set is bounded with respect to the given metric, because all distances are less than or equal to one. So, since boundedness holds vacuously, the problem is equivalent to:
Generally speaking, for a metric space $X$, the following are equivalent:
Indeed, since the whole of $X$ is closed, (i) implies (ii). On the other hand, if $X$ is compact, then all closed subspaces of $X$ are compact, so not (i) implies not (ii).
So, if there are any noncompact closed sets to be found, the whole space will need to be one of them. So at the end of the day the question amounts to just:
This is true. In fact $\mathbb{R}$ with the given metric has the same topology as $\mathbb{R}$ with the standard metric. So, if you know that the standard real line is noncompact, then you see right away that the line with your metric is noncompact too. You could also give an explicit open cover with no finite subcover, such as the union of all balls of radius $1/2$.