Compactness Interpretation?

We define a set $X$ as compact if: "for every open cover of $X$, there exists a finite subcover."

An Open Cover $C$ of $X$ is defined as the union of a collection of Open Sets:
$$C = bigcup_{i in I} A_i$$
and a finite subcover as the union over a finite subcollection:
$$F=bigcup_{i in F} F_i$$

If we define the collection of of open sets to contain only one element e.g. the reals $A={Bbb R}$. Then there exists a finite subcover of $Bbb R$, since we have a finite subcover consisting of only one element i.e. $Bbb R$. Thus, $Bbb R$ is Compact (which is obviously incorrect.)

Take another example: consider any bounded set $E subset Bbb R$. Hence one could construct a superset of E given by: $F supset E $. Let the collection of open sets consist of only element e.g. $A = {F}$. Hence there exists a finite subcover of $E$ consisting of only one element i.e. $F$. Hence $E$ is Compact (regardless of whether or not $E$ is closed or open, which is again incorrect.)

I am struggling to understand the concept of a finite subcover. We could always construct the Open Cover to be the union of a collection of a single set, hence a finite subcover will always exist. Is there a constraint on what a finite subcover should be, other than it should be a finite collection of sets? Do the sets in the collection have to be bounded for example, or is there something else? From what I’ve read, there is no imposed constraint on the finite subcover other than for it to consist of finitely many sets.