Open Cover and Compactness I'm having trouble when thinking about compactness.
We knew that compactness of $A\subseteq\mathbb{R}$, can be defined as:
Every open cover of $A$ has a finite subcover.
My question is:
An open cover is a collection of open sets.
So, when we are talking about the finite subcover, can I have a number of countably infinite  intervals, here?
Because a finite subcover means a finite open sets, and every open set in $\mathbb{R}$ is characterized as disjoint union of a countably open intervals.
Any help will be appreciate.

It might help to write out the definition of compactness in a form commonly used in applications: Let $\Lambda$ be an arbitrary set. If $\{O_\lambda \mid \lambda \in \Lambda\}$ is a collection of open sets indexed by $\Lambda$ such that $A \subset \bigcup_{\lambda \in \Lambda}O_\lambda$, then there exists a finite subset $\{\lambda_1, \dots, \lambda_n\} \subset \Lambda$ ($n \geq 0$) such that $A \subset \bigcup_{i = 1}^{n}O_{\lambda_i}$.
