A topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed into any other such path while preserving the two endpoints in question.
I understand both formal and intuitive meaning of the simple connectedness, but I can't get the idea how they imply each other. For example, given the following 2D-picture of a space that is connected but not simply connected. I've tried to draw two paths that contradict to simple connectedness. I can't understand why we can't build a homeomorphism between them.