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- Feb 7, 2012

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If you have a covering of this space by open sets then one of those sets will contain the point 0 and will therefore contain a basic open neighbourhood of 0, of the form $(-\infty,-n)\cup (-s,s)\cup (n,\infty).$ The complement of this neighbourhood is the compact subset $[-n,-s]\cup [s,n]$ of the usual line, and will therefore have a finite subcover. Therefore the whole covering has a finite subcover. Conclusion: the "looped line space" is compact.Is the looped line space compact ?, the looped line space has the same bases as the usual at R except at 0 it has the form (- infinity , - n ) U ( - s , s ) U ( n , infinity )

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