in R^n, compact is equivalent to closed and bounded, so a closed set is bounded iff compact, and a bounded set is closed iff compact.
in a metric space, compact is equivalent to complete and totally bounded.
in R^n which is itself complete, closed is equivalent to complete, and since every bounded set in R^n has comoact closure, bounded is equivalent to totally bounded.
a totally bolunded set is one in which everys equence ahs a cauchy subsequence, and a completes et one in which everyu cauchy sequence converges.
the connection is that a compact metric space is one in which every sequence has a convergent subsequence. (i think. it has been a long time since i taught this course.)
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