How do closed, bounded, and compact concepts relate in metric spaces?

[Ratzinger]

Could someone explain me how these three concepts hang together?

(When is a set bounded but not closed, closed but not bounded, closed but not compact and so one?)

 

[mathman]

Examples (real line with usual topology).

Bounded not closed: 0<x<1
Closed not bounded or compact: 0<=x<oo.

 

[mathwonk]

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.)