Give example of metric space /normed linear space in which we have a proper subset which is clopen.

Question

In metric space the example of discrete metric space is very trivial as in discrete msp every subset is clased as well as open,now iaaue becomes in case of normed linear space.whether such a normed linear space exists or not?I am unable to find such example please help me to solve this problem.Thanks in adavce.

Answer 1

Let $X$ be a normed linear space and suppose that $A$ is a non-empty proper subset of $X$ which is open and cloded. Let $B:= X \setminus A$. Then $B$ is a non-empty proper subset of $X$ which is also open and closed. Furthermore we have

$(*) \quad X= A \cup B$.

Now take $a \in A$ and $b \in B$ and let $x(t):=a+t(b-a)$ for $t \in [0,1]$. Then $x(t) \in X$ for all $t \in [0,1]$.

It is your turn to get a contradiction to $(*)$.

Answer 2

You need any disconnected metric space! Also note that normed linear spaces are connected, so we cannot find such a set in a normed linear spaces!

For example, In $\Bbb{Q}$ with the usual metric, the set $(-\sqrt{2},\sqrt{2}) \cap \Bbb{Q}$ is clopen