I just found a cool metric space in measure theory, what is your favorite metric/topological/functional space?

Hey guys, so I've recently started teaching myself measure theory and Lebesgue integration, and I came across a really cool (in my opinion) metric space. It is problem 12, chapter 1 in Folland. I'll set it up for y'all.

Let (X,M,m) be a finite measure space. We define the equivalence relation ~ as follows: For E,F in M, if m(E[;\Delta;]F)=0, then EF. E[;\Delta;]F is the symmetric difference, i.e., E[;\Delta;]F= E\F U F\E. It is fairly easy to see that this does define an equivalence relation on M. So, the metric space is as follows: For E,F in M, let p(E,F)=m(E[;\Delta;]F). This creates a pseudometric, but a regular metric arises from E,F in M/ where d(E,F)=m(E[;\Delta;]F).

I dunno why, but I was really intrigued and happy to see that this does indeed a metric space. Last year, I had two classes on metric space theory, topology, and intro functional analysis, so it was neat to see that stuff come up again. I ask you math redditors, what are some cool spaces that are relatively simple (I don't think metric spaces are that advanced) that have came up when you later did some more advanced work?

*Edit: formatting. Anyone know how to correctly write the symmetric difference here?

**Edit 2: fixed prior formatting mistakes!