What’s so special about [0,1]?

It happens to everyone. First, you learn real analysis and suddenly you understand what it means to do rigorous mathematics. You learn the precise definition of what a continuous function is; you learn that if they are defined on a closed bounded interval then they are bounded and attain their bounds; you learn about things […]

What’s the difference between an orientation and a rotation?

This question came up at work recently, and I was unable to find a really good answer online. For the purposes of this article, I will be using the terms orientation and rotation as they are used in computer graphics; this might not be the same definitions you are used to. If you search online […]

The topological closed graph theorem

There are a number of theorems in various settings that link continuity of a function \(f\colon X \to Y\) to closedness of that function’s graph – i.e., the set of all points \((x,y)\in X \times Y\) such that \(y = f(x)\). For example, we learn in functional analysis that if \(f \colon X \to Y\) […]