A while ago, I wrote about the importance of open sets in topology and how the properties of a topological space $X$ are highly dependent on these special sets. In that post, we discovered that the real line $\mathbb{R}$ can either be compact or non-compact, depending on which topological glasses we choose to view $\mathbb{R}$ with. Today, I’d like to show you another such example - one which has a surprising consequence!

Last time we proved that the pseudo-hyperbolic metric on the unit disc in ℂ is indeed a metric. In today’s post, we use this fact to verify Lindelöf’s inequality which says, "Hey! Want to apply Schwarz's Lemma but don't know if your function fixes the origin? Here's what you do know...."

In this post, we define the pseudo-hyperbolic metric on the unit disc in ℂ and prove it does indeed satisfy the conditions of a metric.

Love and Math by Edward Frenkel is an excellent book about the hidden beauty and elegance of mathematics. It is primarily about Frenkel’s work on the Langlands Program (a sort of grand unified theory of mathematics) and its recent connections to quantum physics. Yet the author's goal is not merely to inform but rather to convert the reader into a lover of math. While Frenkel acknowledges that many view mathematics as an “insufferable torment… pure torture, or a nightmare that turns them off,” he also feels that math is “too precious to be given away to the ‘initiated few.’” In the preface he writes...

The goal of today's post is to prove that the fundamental group of the real projective plane, is isomorphic to $\mathbb{Z}/2\mathbb{Z}$ And unlike our proof for the fundamental group of the circle, today's proof is fairly short, thanks to the van Kampen theorem! To make our application of the theorem a little easier, we start with a simple observation: projective plane - disk = Möbius strip. Below is an excellent animation which captures this quite clearly....