Please give me examples of disconnected metric spaces that cannot be isometrically embedded into connected metric spaces.   I have just started to study connected metric spaces and I haven't found out a metric space $X$ which isn't a subspace from another metric space such that $X$ isn't connected.  Every counterexample that I've looked, has been subspaces from a bigger metric space, e.g. $[0, 1]\cup[2, 3]$ which fails to be an interval in $(\mathbb{R}, |\cdot|$).   It might be topological space that isn't connected but for metric spaces? is there any apart from discrete metric spaces?  Thanks in advance
Is every metric space a subspace of some connected metric space? If the space itself is connected then we're done, but if not then I think we can extend our metric space to make it connected.  I'm not sure whether this will work or not, but intuitively I think the answer is yes.
Is every metric space a subspace of some connected metric space? If the space itself is connected then we're done, but if not then I think we can extend our metric space to make it connected.  I'm not sure whether this will work or not, but intuitively I think the answer is yes.