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Have Examples and Counterexamples

Date Written: June 18, 2024; Last Modified: June 18, 2024

Thanks to my office mate Harahm for suggesting this!

On exams, it’s not uncommon to mostly solve a problem and say, “Hmm, statement X would finish it off, but I don’t know if it’s true”. Perhaps you are tempted to say “In a metric space, a closed ball of radius \(1\) is always compact”.

This is somewhat difficult to disprove in an exam environment, especially if there is pressure to finish the exam on time and get to solving other problems. In situations like this, having a million different counterexamples for different things is extremely helpful.

Let’s continue with this example — likely, one would review the following theorem while studying:

One may be tempted to apply this theorem in a general setting and affirm, “Yes, the closed ball of radius \(1\) is always closed and bounded, even in a non-Euclidean metric space.”

(By the way, this is another situation where simply knowing the theorem isn’t enough: knowing that Heine-Borel relies on the fact that bounded subsets of \(\mathbb{R}^n\) can be covered by finitely many \(\epsilon \)-balls is critical to the proof, and this property does not generalise.)

Of course, Heine-Borel does not apply to general metric spaces; a simple counterexample is the trivial metric on any infinite metric space.

The point here is that knowing Heine-Borel and even knowing the proof intimately does not immediately resolve the truth value of our statement, but having a counterexample to the theorem when \(\mathbb{R}^n\) is replaced by an arbitrary metric space very quickly resolves it.

I will say that such scenarios arise frequently in exams. As someone that has sat through many exams and graded many more, there are a lot of statements that seem plausible and are difficult to evaluate on the spot. Being prepared with examples and counterexamples to all sorts of theorems, statements, and definitions is invaluable for these.