# Taking off the vest while the jacket is still on

My topology professor said that:

One can take off his vest while the jacket is still on.

Is there a topological answer to this question?

Of course there are some ground assumptions like :
1- Jacket is on top of vest.
2- Tearing is not allowed.
3- One does not simply break his arms just to take off his vest.
4- Vest and jacket are worn properly.

• slide your arms to your sides, out of the jacket sleeves and underneath the vest. Then, pull the vest out of the neck hole, or slide it down like you're taking off pants. – mojambo Jan 5 '14 at 14:31
• One needs a pretty elastic vest for that, if one doesn't take one's arms out of the jacket sleeves. But it works topologically. – Daniel Fischer Jan 5 '14 at 14:32
• Stuck your whole body + feet + head in the left arm hole of vest, then pull the vest out of the right sleeve of jacket? – peterwhy Jan 5 '14 at 14:38
• There is a Youtube video of this. – Old John Jan 5 '14 at 14:49
• Equivalently take off your underwear without removing your trousers.One mathematician used to actually do this for an audience. There is an episode of Mr.Bean (Rowan Atkinson) doing this in reverse : Putting his swimming briefs on, under his trousers, without removing his trousers. – DanielWainfleet Nov 16 '15 at 23:08

The topological statement is "There is a relative homotopy of the three-dimensional situation where the vest is worn to the situation where the vest is removed, relative to the body and the jacket." As mentioned in the comments, the vest may need to be sufficiently elastic. Also, popular demonstrations of the property cheat by moving the body and jacket during the procedure.

Interestingly, many popular explanation about what topology is are about homotopy rather than homeomorphism. Often one even reads that a coffee cup and a donut are homeomorhic ecaxtly because(!) they can be transformed into each other without tearing or glueing when made out of rubber. Of course this property is much stronger than homeomorphism, it is about homotopy of embeddings into $\mathbb R^3$.

• I believe the property is even stronger, because we require that the image of the vest be homeomorphic to a vest throughout the homotopy, and so we're in fact looking at an isotopy rather than just a homotopy. – Dan Rust Jan 8 '14 at 14:27