Axiom of Choice in General Topology

Tychonoff's theorem, which states that the product of any set of compact topological spaces is compact with respect to the product topology, is equivalent to the Axiom of Choice. And Tychonoff's theorem is one of the most important theorems in Topology.

See the papers "Horrors of topology without AC, a non-normal orderable space" (van Douwen) and "continuing horrors of topology without choice" (Good and Tree)

Basic things that can go wrong in the absence of choice: $\mathbb{R}$ can be the union of countably many countable sets (so is meagre in itself and Baire's theorem fails), a sequentially continuous function on a metric space need not be continuous, etc etc. It's used in very many places, especially its countable form. We cannot define compactifications in the usual way, a lot of dimension theory becomes invalid, no nice theory of ordered spaces etc. It's certainly not confined to just covering properties; it touches almost all parts of topology and analysis.