Let me venture into the realm of σ-algebras. Yes, you might say, that is measure theory, not topology… but topology plays an important role in measure theory and, for that matter, descriptive set theory. I will tell you about sets with the Baire property. Those are pretty simple objects, or at least they appear to be simpler than Borel sets, but we will see that this is the other way around: all Borel sets have the property of Baire. The proof is pretty easy, as well. I will also spend some more time to explain a more complicated result, due to O.M. Nikodým, and which says that all A-sets have the property of Baire as well, in a second-countable space. None of that ever uses any Hausdorffness, or in fact any separation property whatsoever. Read the full post.