Here's a given definition while I am reading the Topology, Munkres.
Definition. A space $X$ is said to be compact if every open covering $A$ of $X$ contains a finite subcollection that also covers $X$.
Then my question here, when we check whether the open covering $A of X contains a finite sub collection that also covers X, that sub-collection must be proper or not?
I think it must be proper, or not that I haven't known that sub means equal to proper seeing the next example :
The interval $(0, 1]$ is not compact; the open covering $A = \{(1/n, 1] | n ∈ Z^+\}$ contains no finite subcollection covering (0, 1].