My question is related with the understanding of open covering definition of compactness which can be stated as: A metric space $X$ is called compact iff each of its open cover has a finite subcover. For example $[0,1)$ is not compact. I understand what open covers are; for example set $A = \{(-1,1)\}$ may be one of open coverings of $[0,1)$. But I don't understand how to check whether set A admits a finite subcover or not? I tried many times but I don't know where I am lacking? Please help me with this.
Edit 1: Let $O$ be the set of open intervals of the form $(-1/2, 1-1/t)$. $O$ is an open cover but it does not admit a finite subcover. My question is related with understanding of how can we say $O$ doesn't admit a finite subcover.
Edit 2: I find it difficult to apply the open covering definition of compactness. If someone can explain this definition with examples it would be good for me.
Thanks