These are three questions about metric spaces with the intrinsic metric on them.
Let $(M,d)$ be a length metric space, i.e. its metric is the intrinsic metric.
Can such a space have a point $p$ whose distance $d(p,q)$ from all other $q \in M$ is not less that some $s > 0$, i.e . a discrete point $p$, which can be isolated by an open ball $B_s(p)$ with $B_s(p) \cap M = \emptyset$ ?
Does every open ball $B_s(p)$ at $p$ contain a point $q \neq p$ ?
Can every two points $p,q \in M$ be connected by continuous curve $\gamma : [0,1] \to M$, $\gamma(0) = p, \gamma(1) = q$ ?
The answers to the first and second seem to be no and yes, respectively, because of the property of having approximate midopints (see the Wiki article above). Could anyone confirm and elaborate.