I just need help understanding this problem:

Suppose that is a family of topological spaces and suppose that and are members of . If and differ in only one coordinate, prove that they must lie in the same connected component.

What we have to work with:

Let and be topological spaces. If is connected and the function is continuous relative to and , then is connected as a subspace of .

Suppose that is a family of topological spaces. Fix ; and for all , fix . Define the function by , where and when . Using this map, it can be shown that is homeomorphic to the subspace .

Both of these have been proven in class. I'm just failing to see how to use these when one coordinate differs. Any help is greatly appreciated!