Suppose $X$ is a metric space.Then $X$ is complete iff For each sequence of non-empty closed sets viz. $(F_n)$such that $F_{n+1}\subset F_n$ and $\operatorname {diam}(F_n)\to 0$,$\cap_{n=1}^{\infty} F_n$ is a singleton.
There is another similar property,$X$ is compact iff For each sequence of non-empty closed sets viz $(F_n)$ such that $F_{n+1}\subset F_n,\cap_n F_n$ is non-empty.
I am confused with the nomenclature of these two properties,which one is called Cantor intersection property and what is the other one called?