Show that S = [0,1) is not compact by giving an closed cover of S that has no finite subcover.
The Attempt at a Solution
I know that S is not compact because it is [STRIKE]an open[/STRIKE] not a closed set even though it is bounded.
But I am completely lost on the open cover part.
I understand an open cover is a union of open sets where S is a subset of the union...
But I appear to missing something very fundamental. If I picked (-1, 2) for the cover that is an open set and S is a subset of it's "trivial" union. Why doesn't that work?
<edit> OK, I understand it needs to work for every cover, that's why,
but is (-1,2) a cover? <end edit>
If instead I had [0,1], that is closed and bounded so it is compact.
What sort of sets would go into a cover for it?
Does anyone know some extremely elementary references for this?