It's possible that a set $X$ can be endowed with two or more topologies that are comparable. Over the years, mathematicians have used various words to describe the comparison: a topology $\tau_1$ is said to be coarser than another topology $\tau_2$, and we write $\tau_1\subseteq\tau_2$, if every open set in $\tau_1$ is also an open set in $\tau_2$. In this scenario, we also say $\tau_2$ is finer than $\tau_1$. But other folks like to replace "coarser" by "smaller" and "finer" by "larger." Still others prefer to use "weaker" and "stronger." But how can we keep track of all of this?
Resources for Intro-Level Graduate Courses
In recent months, several of you have asked me to recommend resources for various subjects in mathematics. Well, folks, here it is! I've finally rounded up a collection of books, PDFs, videos, and websites that I found helpful while studying for my intro-level graduate courses.
A Ramble About Qualifying Exams
Today I'm talking about about qualifying exams! But no, I won't be dishing out advice on preparing for these exams. Tons of excellent advice is readily available online, so I'm not sure I can contribute much that isn't already out there. However, it's that very web-search that has prompted me to write this post.
You see, before I started graduate school I had heard of these rites-of-passage called the qualifying exams.* And to be frank, I thought they sounded terrifying.