Question

Show that every compact subspace of a metric space is bounded in that metric and is closed. Find a metricspace in which not every closed bounded subspace is compact?

Answer 1

Answer:

If M itself has this property, then we say that M is a compact metric space. We start with the fact that in any metric space, a compact subset is closed and bounded. Bounded here means that the subset “does not extend to infinity,” that is, that it is contained in some open ball around some point.