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.
The interval [0,1] is compact because every infinite subset of it has a limit point that remains inside [0,1]. For example, the subset {1/n} accumulates at 0, and since the interval is closed, 0 is included in [0,1].
Conditions: E is the closed interval [0,1].; Compactness is judged by Hausdorff's limit-point definition.; S is an infinite subset of [0,1].
The interval [0,1] is compact because every infinite subset of it has a limit point that remains inside [0,1]. For example, the subset {1/n} accumulates at 0, and since the interval is closed, 0 is included in [0,1].
Conditions: E is the closed interval [0,1].; Compactness is judged by Hausdorff's limit-point definition.; S is an infinite subset of [0,1].