Wrong question!

The only set which is intrinsically closed or open is the empty set (which is both). Every other set may be open, closed, both, or neither, depending on what topological space it is embedded in.

If the set of extended real numbers is considered as a topological space in its own right then (as is the case with the "whole set" of any topologocal space) it is both closed and open - in that space.

Good Hunting!
RD