A connected set is a set that cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other. Let X be a topological space. A connected set in X is a set A⊆X which cannot be partitioned into two nonempty subsets which are open in the relative topology induced on the set A. Equivalently, it is a set which cannot be partitioned into two nonempty subsets such that each subset has no points in common with the set closure of the other. The space X is a connected topological space if it is a connected subset of itself.
closed set | connected space | empty set | open set | set | set closure | simply connected | subset
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