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 denotes a manifold of dimension with an associated differential structure.

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