Path connectedness of union of path connected spaces

[radou]

Homework Statement

As the title suggests, Let {Aj} be a collection of path connected subspaces of some space X, and let the intersection of these subspaces be nonempty. Is U Aj path connected?

The Attempt at a Solution

Again, my answer would be no, in general.

But, since their intersection is nonempty, that means that all these subspaces share at least one element in common. So, if we choose any x from U Aj, and any y from the intersection, we can find a path which connects them, since x is in Aj, for some j, and y is in Aj, too.