[sdenton4]

Misunderstanding also exists, though, at many stages of the game. There's a pair of famous books - counterexamples in topology and another on analysis - dedicated entirely to counterexamples which dispel some common mis understandings.

And these misunderstanding are not limited to undergrads; often they stand in the way of progress at the edge off research, simply because we don't actually know what's possible. Research level math is often guided by folklore: important conjectures and shadows of what might be. And when the folklore is wrong, we go down the wrong track until someone finds a counterexample...