Here is a list of some of the subsets of integral domains, along with the reasoning (a.k.a proofs) of why the bullseye below looks the way it does. Part 2 of this post will include back-pocket examples/non-examples of each.   

The notion of a topological space being Hausdorff or normal identifies the degree to which points or sets can be "separated." In a Hausdorff space, it's guaranteed that if you pick any two distinct points in the space -- say $x$ and $y$ -- then you can always find an open set containing $x$ and an open set containing $y$ such that those two sets don't overlap.

After some exposure to group theory, you quickly learn that when trying to prove a group $G$ is abelian, checking if $xy=yx$ for arbitrary $x,y$ in $G$ is not always the most efficient - or helpful! - tactic. Here is a (not comprehensive) running tab of other ways you may be able to prove your group is abelian:

Yes! I'm writing about math. No! Don't close your browser window. Hear me out first...

I know very well that math has a bad rap. It's often taught or thought of as a dry, intimidating, unapproachable, completely boring, who-in-their-right-mind-would-want-to-think-about-this-on-purpose kind of subject. I get it. Math was the last thing on earth I thought I'd study. Seriously.