In the midst of learning about compactness I come across Tychonoff's Theorem:
Let $\{X_i : i \in \mathcal{A}\}$ be any collection of compact spaces. Then $\displaystyle\prod_{i \in \mathcal{A}}X_i$ is compact in the product topology.
I've just come from the fact that a finite product of compact spaces is compact, and I also know from studying bases of topologies that uncountable products aren't necessarily as nice (for example, the box topology has some problems for uncountable products).
The proof for Tychonoff's Theorem is:
Omitted (this is much harder than anything we have done here).
Internet searches lead to math overflow and topics that are very outside of my comfort zone.
Is there a proof of Tychonoff's Theorem for an undergrad?
Perhaps the most elementary proof is the one that I first encountered as a freshman, using the Alexander subbase lemma. It requires Zorn’s lemma, but it does not require knowledge of filters, ultrafilters or nets. It’s carried out completely in this PDF. (And Alexander’s result is of some interest in its own right.)