owlpride wrote:Back up a second. What's your definition of compactness? Judging from your approach, I assume it's "a space is compact if any open cover has a finite subcover"?
Yes, you can cover (0,1) by itself. Or by (0,0.55) union (0.45,1). Both are finite open covers, but that does not tell you anything about the compactness of (0,1).
What you need to be worried about are infinite open covers. If a space is compact, any and every infinite open cover admits a finite subcover. So let's look at the open cover An=(1/n,1-1/n). Does it admit a finite subcover? If not, (0,1) is not compact.
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