Connectedness

Connectedness is transitive

Let be a topological space and . If is (path-)connected to and is (path-)connected to , then is (path-)connected to . topology Hence both forms of connectedness form an equivalence relation.

For plain connectedness

Let and , i.e. there exists connected subspaces and . We claim that is a connected subspace of . Let denote that natural inclusions of in , and be a continuous function. Since and are continuous, so too are . Thus for all . Hence is constant for . Therefore .

For path connectedness

Let be a continuous path from to and be a continuous path from to . Then the product is a continuous path from to . Hence .

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