Let be a topological space and .
If is (path-)connected to and is (path-)connected to ,
then is (path-)connected to . #m/thm/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 .