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


*

*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 
 A: 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?
A: It sounds like in the book you're thinking about $F$ as a subset of $\mathbb{R}$. So then you can also consider the set $U$ as you've defined, since each element $I \in U$ is an open subset of $\mathbb{R}$ and $U$ covers $F$.
Of course, if you wanted to think of $F$ as the whole topological space you're working with (i.e., not as a subset of $\mathbb{R}$), you can take the open cover $U$ of $F$ from $\mathbb{R}$ and intersect its elements with $F$. In other words, we can consider $U' = \{I \cap F : I \in U\} = \{\lbrack 0, n) :  n\in\mathbb{N}\}$. It's immediate (and easy to see directly) that this is an open cover of $F$ in the subspace topology.
