# Topological Magic: Infinitely Many Primes

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!

# The Pseudo-Hyperbolic Metric and Lindelöf's Inequality (cont.)

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...."

# The Pseudo-Hyperbolic Metric and Lindelöf's Inequality

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.

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....