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.
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.