Miths V: Topologists

As Grisha made news for solving the famous conjecture and subsequently refusing the Fields Medal, various media had opportunities to characterize topology among the branches of mathematics. Here's the NYT snapshot:

The conjecture is fundamental to topology, the branch of math that deals with shapes, sometimes described as geometry without the details. To a topologist, a sphere, a cigar and a rabbit's head are all the same because they can be deformed into one another. Likewise, a coffee mug and a doughnut are also the same because each has one hole, but they are not equivalent to a sphere.

You might recall the article had two images: one of Grisha's face morphing into a sphere, and another of a rabbit decorated with latitudinal and longitudinal lines indicating how it might be deformed into a sphere. Topology is often called “rubber geometry” because two objects are topologically indistinguishable if (roughly) a rubber version of one could be transformed without tearing into the other. So in particular, spheres of different radii are the same topologically. They are different geometrically since the sphere of larger radius is “flatter,” in a sense that can be made precise.

Okay, great. Let's be a little bit pedantic.

To a topologist, a sphere, a cigar and a rabbit's head are all the same.

Properly, a sphere, a cigar and a rabbit's head are all the same as topological spaces. Who knows about the mental states of those who study topology, and anyway why would those mental states carry over to the daytime topologist's night- or family-life?

The World According to a Topologist, According to an AS reader of the above Times article

Topologist at the day care center
Topologist: I'm here to pick up my child.
Supervisor: Okay, she's napping now. I'll go wake her up.
Top: No, let her sleep. I'll take another child.
Sup: Excuse me?
Top: They're topologically indistinguishable!!

Topologist at the bike shop
Top: Do you have any deals on used bikes?
Hippie Bike Shop Dude: Well, we have a great Schwinn, but one of the wheels was bent out of shape when the last owner ran over the bike with his car.
Top: Did the wheel break into two pieces?
HBSD: No.
Top: Hm, okay. Did the rim snap at any point?
HBSD: Nope, it's just bent, man. P.S. I am high.
Top: Okay, I'll take it.
HBSD: Do you want us to fix it, or do you have some expertise yourself?
Top: I'm no mechanic, but it's fine. The accident did not affect the bike topologically.

Topologist on trial
DA: So you admit to stabbing the victim four times in the chest?
Top: Yes.
DA: Ummm, okay.
Top: I never stabbed all the way through, so I didn't change the victim's homeomorphism type!! Not guilty, y'all's got to feel me.

Topologist after receiving the first transplant of its kind
Doctor: Okay, you're all set.
Top: Take it off.
Doc: I'm sorry, why?
Top: You topologically altered the braindead guy to obtain this organ.
Doc: Really, that's it? You understand you're quite lucky to've received this transplant. Your reason is a little bizarre.
Top: Also it is freaking me out that I have to touch this thing every time I pee.