Consider the following two definitions:
Connected : A topologiocal space X is connected if it is not the disjoint union of two open subsets, i.e. if X is a disjoint union of two open sets A and B, then A or B is empty set.
Path Connected : A topological space X is path-connected if any two points in X can be joined by a continuous path.
So far I can picture, I think they should be equivalent. If X is connected iff it is path-connected. But, the Thm I got from lecture is just if X path connected then X is connected (The converse not necessarily true.)
Can anyone give me some picture of space which is connected but not path-connected? Just picture please, cuz I wanna grab the idea. I have already seen some examples like in connected but not path connected?, I cannot grab the idea if it (such space) consist of a single piece, then I couldn't make a continuous path from any two points.
Thank you :)