So it’s cold and rainy, and you’re up a little too late trying to figure out why that one pesky assumption is necessary in a theorem. Wouldn’t it be nice if you could just order up a space that was path connected but not locally connected? You’re in luck, there’s
an app a website for that. The π-Base will deliver an Alexandroff Square right to your door computer screen.
The π-Base is a souped-up version of Steen and Seebach's book Counterexamples in Topology that anyone can contribute to. As the name suggests, Counterexamples in Topology is filled with unusual topological spaces and the topological properties they do and don't have. A counterexample is a concrete example that shows us that a particular general statement is not true. For the statement "all odd numbers are prime," 9 is the first counterexample. In topology, counterexamples are spaces that have one property but not another, showing that the two properties don't have to coexist. Many of the spaces in Counterexamples in Topology are truly bizarre. Normally when we think of topology, we think of the whole "donut into coffee cup" thing, where we have two spaces that we can make out of Play-Doh and morph into each other. The counterexamples in the book and on the π-Base are much harder to visualize. For example, check out Cantor's Leaky Tent (which must have the best name on the whole site). This space has no line segments in it, so there's no way to make it out of Play-Doh and mush it around to see what happens.
James Dabbs, who started the π-Base, was a grad student in topology at Auburn working with cozero complement spaces. (Don't worry, I didn't know what those were either, until I started writing this post. Dabbs' advisor Gary Gruenhenge has a research article about them here.) This property was not in Counterexamples in Topology, and Dabbs was stuck with "a stultifying literature search" just to find examples and determine what other topological properties they had. Austin Mohr, a mathematician at Nebraska Wesleyan University, a similar feeling. "I used to be fairly active on Math Stackexchange, and I found myself answering questions like 'Are there any spaces that have property x and y, but not z?'. I always went to Counterexamples in Topology for such questions, but the reference chart is cumbersome to use if many properties are involved," he wrote in an email. He put together a site called Spacebook that was basically a searchable version of Counterexamples in Topology, but when he saw that Dabbs was working on a more extensive project, he shifted his efforts to π-Base.
It might seem strange to expend so much effort on finding and describing counterexamples, but there are a lot of good reasons why they're important in any field of math. First, they're fun. Whether you're finding functions that are continuous but not differentiable (Weierstrass' "monsters") or trying to understand what the heck Cantor's Leaky Tent is, they will amuse you and stretch your brain.
But they're also important for practical (or at least as practical as anything else in theoretical math) reasons. We want to prove theorems that are as general as possible, and counterexamples can help us check to see whether all our assumptions are necessary. Counterexamples can also help us find proofs. If we're trying to prove a theorem with certain assumptions, it helps to know why the theorem won't work in other circumstances. If we can compare what the theorem would say about two different spaces with slightly different properties, we can get a better idea of what goes wrong when we take an assumption away. Conversely, counterexamples can also help give us a reality check about a proof. If you've proved a theorem, but your proof works for a space it shouldn't work for, you've done something wrong. For all of these reasons, counterexamples are pedagogically valuable (although it is possible to go overboard—a topology class that just consists of lists of counterexamples is not ideal). Counterexamples help students understand the limits of their intuition.
The π-Base is more than just a searchable online version of Counterexamples in Topology. In addition to listing topological spaces and their properties, the program can actually prove theorems about those spaces and properties. If every space that has properties A and B also has property C, then the π-Base will add property C to all spaces with properties A and B. This means people don't have to enter each property manually, and the site can find properties people haven't yet bothered to check or record. Currently, the site is limited to simple deductions, but Dabbs hopes to incorporate an automated theorem assistant like Coq or Isabelle to prove more complicated theorems. He has also suggested that perhaps the program could "mine" for conjectures by finding combinations of properties that no spaces have, so it would be both a theorem-suggesting and a theorem-proving program.
But for now, people are doing most of the heavy lifting, and you can help! Mathematicians can add spaces, properties, and proofs to the site. Programmers can get the source code at Github and work on some of the bugs and desired features. You can even click the "Contribute" button at the top of the page to get a random assertion that needs a proof. I'm teaching topology right now, and it's tempting to assign those to my students! Mohr has used the π-Base in undergraduate research courses and says that replacing the computer's automated deductions with human-enlightening proofs would be valuable as well, especially for students. The more we contribute, the more useful it will be for all of us.