-Professor, why is Urysohn's lemma important?-
-Why is this important? ... ... What is the meaning of important? Is math important?... baah, I do not know. But I hope it is at least interesting.- B. Zimmermann
I strongly believe that all comes from the non trivial fact that 4 is 2+2 but also it is 2x2. Moreover I refuse to think that 2^2 is playing any important role here.
My favorite connected subspace of the plane that is not path-connected.