Gelbaum-Olmsted <i>Counterexamples in analysis</i> is a standard reference for various pathological examples in analysis:<p><a href="https://archive.org/details/counterexamplesi0000gelb" rel="nofollow">https://archive.org/details/counterexamplesi0000gelb</a>
Cantor's set always messes with my mind.<p>Say you start with the standard construction: take closed interval [0, 1], then remove open middle third, then remove two open middle thirds from two remaining segments, then remove 4 open middle thirds from 4 remaining segments, etc countably many times (more formally, intersect all of the sets you obtain in each of the step).<p>Intuitively, what remains should be the end points of the removed intervals, but this is not true: since at every step you removed finitely many intervals, and there were countably many steps, you removed finite<i>countable <= countable</i>countable = countably many intervals, so there are only countably many many ends of removed intervals, but the Cantor's set is uncountable. So where do the remaining points come from?
> <i>Things fall apart when the space is not compact. Your sequences may converge but not to an element in your space.</i><p>This is about closedness only. ℝ is not compact but contains its limit points.