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Removing the endpoint 0 makes (0,1] noncompact because the infinite subset {$1/n$} still lies inside (0,1], but its limit point 0 is no longer included in the set. This violates the limit-point compactness condition that every infinite subset must have a limit point within the set.

Conditions: E is the half-open interval (0,1].; S is the infinite subset {$1/n$ : $n=1$,2,3,...}.; Compactness is judged by the limit-point definition.

Removing the endpoint 0 makes (0,1] noncompact because the infinite subset {$1/n$} still lies inside (0,1], but its limit point 0 is no longer included in the set. This violates the limit-point compactness condition that every infinite subset must have a limit point within the set.

Conditions: E is the half-open interval (0,1].; S is the infinite subset {$1/n$ : $n=1$,2,3,...}.; Compactness is judged by the limit-point definition.