Compactness of $[0,+\infty)$ Let's say we have $$F = [0,∞).$$ 


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*How can we determine whether this is a compact set or not?

*And let's say we have $U = {(-1,n)}$ ($n∈N$), book said that this $U$ is the open cover of $F$, but I thought $U$ is not a subset of $F$, so it can not be an open subset of $F$?


Correct my thoughts please. 
thank you 

Open cover of space $F$ is a family $\mathcal U = \{U_i\}_{i\in\mathbb I}$ of open sets such that $\bigcup_{i\in\mathbb I}U_i = F$. So $\mathcal U$ is not a subset of $F$, but every element of $\mathcal U$ is. To show that a space is not compact it is enough to find open cover for which no finite subcover exists. Just consider cover given in (2). Is there open subcover of it?
