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].
Hausdorff defines compactness by requiring that every infinite subset S of a set E has a limit point that lies within E itself. This prevents infinite subsets from escaping to infinity without accumulating inside the set.
Conditions: E is a set.; S is an arbitrary infinite subset of E (S ⊂ E).; The limit point must belong to E.
Hausdorff defines compactness by requiring that every infinite subset S of a set E has a limit point that lies within E itself. This prevents infinite subsets from escaping to infinity without accumulating inside the set.
Conditions: E is a set.; S is an arbitrary infinite subset of E (S ⊂ E).; The limit point must belong to E.