Blogging Counterexamples

I can’t believe someone has been blogging about counterexamples since July of last year and I just found out! Luckily, the Aperiodical Advent Calendar alerted me to it yesterday, and now Math Counterexamples is the newest addition to my blog feed.

The Smith-Volterra, or “fat” Cantor set, a great counterexample in topology. Image: Inductiveload, via Wikimedia Commons.

Counterexamples have been on my mind a lot lately. This spring, inspired by a post I wrote about the π-Base, an online analogue of Counterexamples in Topology, I started writing about some of my favorite spaces at my Scientific American blog Roots of Unity. Many of these spaces are counterexamples: connected but not path connected, nowhere dense but having positive measure, contractible but in no obvious way. While my posts tend to be focused on geometry and topology, Jean-Pierre Merx, who writes Math Counterexamples, is an equal-opportunity counterexample-ist. His posts cover topics in algebra and analysis as well as topology.

Like Tai-Danae Bradley’s blog Math3ma, which I wrote about a couple months ago, I imagine Math Counterexamples would be a great resource for undergraduate or graduate students in math. (Their professors should probably pay attention, too: these are great resources to share with your students, but you may also want to make sure the questions you give aren’t immediately google-able!)

I’m especially partial to the analysis counterexamples right now because they’re useful but I often have trouble cooking them up on my own. A recent post that caught my eye is about a function from R² to R with a local minimum at (0,0) on all lines through the origin but which doesn’t have a local minimum there when considered as a function of two variables. Weird, huh? If you go poking around the archives, I’m sure you’ll find some fun factoids too.