This is an exercise in W. Rudin's book. Actually my question is, what is an open cover of a metric space? Since an open set is embedded in a certain metric space, how can it cover those points which lie on the bound of the metric space? Should we define a "larger" metric space?
I have read about compactness of metric spaces on Wikipedia. It says:
A metric space M is compact if every sequence in M has a subsequence that converges to a point in M. This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness defined via open covers.
And I'm still confused about it.