6
votes
2answers
130 views

Closed, orientable surface whose genus is very hard to find intuitively

I'm introducing the Classification Theorem on closed and orientable surfaces in a talk on (intuitive) topology, and to motivate it I'd like an example of an embedding of a surface in $\mathbb{R}^3$ ...
4
votes
5answers
424 views

Motivation for the importance of topology

Starting from tomorrow, I will be tutoring some undergraduate students following a course in general topology. I am looking for examples motivating the importance of topology in mathematics which can ...
2
votes
1answer
430 views

Compact and Locally Compact Spaces

I would like to consult with anyone who is reading this post on how do you explain the distinction between compact spaces and locally compact spaces to students who had just completed topology course ...
19
votes
8answers
3k views

Why do introductory real analysis courses teach bottom up?

A big part of introductory real analysis courses is getting intuition for the $\epsilon-\delta$ proofs. For example, these types of proofs come up a lot when studying differentiation, continuity, and ...
2
votes
1answer
106 views

weakly locally one-to-one?

Is there any standard name for this concept that is weaker than local one-to-one-ness? In some open neighborhood of $x_0$ there is no point $x\ne x_0$ such that $f(x)=f(x_0)$. Or, if you like: In ...
28
votes
7answers
2k views

Quotient geometries known in popular culture, such as “flat torus = Asteroids video game”

In answering a question I mentioned the Asteroids video game as an example -- at one time, the canonical example -- of a locally flat geometry that is globally different from the Euclidean plane. It ...