compact sets 2

We shall summarize the definitions of compact sets.
At first glance, you may not think these are equivalent.
However, these present a same concept and show the properties of compact sets.

(1)A set is compact if and only if any open cover of the set has a finite open sub cover.

It  depends on the theorem of Heine-Borel. Please note "finite".

(2)A set in $\mathbb{R}$  is compact if and only if the set is complete and bounded.

Hence, a closed interval in $\mathbb{R}$  is compact. It is understood easily.

(3)A metric space is compact if and only if any infinite sequences has a convergent sub sequence.

It is called the theorem of Bolzano-Weierstrass.

Note that these theorems would not be true if the existing conditions were changed even if only slightly.