From the definition of a compact set, a set $A$ is said to be compact if for every open cover of $A$, there exists a finite subcover such that $A$ is a subset of this finite subcover.
Here if we consider the open cover, is it necessary for all the elements (which are sets) of this open cover to be of finite?
Also should the finite subcover have elements which are finite?
If we consider the subcover to have the set $(-\infty, +\infty)$ as one of its sets, then we have that all sets of $R$ compact.
But while proving theorems related to compact sets (from Rudin), this property of finiteness of the sets is never used. So should the elements of the finite open cover be finite or that we should not include the set $(-\infty, +\infty)$ in specific.