# One week in

So, I’ve officially survived the first week of my (second) second year as a Ph.D. student. It’s hard to imagine sometimes that I’ve been a grad student for over three years.

I’m such an old man.

It’s been even less easy than I’d anticipated, too, which is nicht spaß; sadly, too, this is the first week where teaching responsibilities are actually a thing (we get week 1 off from such things), so now it’s difficult and time-consuming. Jajajajaja.

Rather than making this entry all about `Yours Truly`, I figured I’d pit stop in and use some of my (*gasp*) unoccupied time write up a little spiel I saw for the first time in my Riemannian geometry class last week.

In topology, there’s an idea called invariance of dimension which can be stated in many different contexts, situations, etc. This can be modified in the case of manifolds with differential structures, and because the idea of the proof seemed a bit cool, I decided to throw it up here for you guys. Throughout, the notation $M^m$ denotes a manifold $M$ of dimension $m$ with an associated differential structure.