Editing Homotopy (section)

=== Homotopy equivalence vs. homeomorphism ===
A [[homeomorphism]] is a special case of a homotopy equivalence, in which {{nowrap|1=''g''&thinsp;∘&thinsp;''f''}} is equal to the identity map id<sub>''X''</sub> (not only homotopic to it), and {{nowrap|1=''f''&thinsp;∘&thinsp;''g''}} is equal to id<sub>''Y''</sub>.<ref>Archived at [https://ghostarchive.org/varchive/youtube/20211211/XxFGokyYo6g Ghostarchive]{{cbignore}} and the [https://web.archive.org/web/20200829013025/https://www.youtube.com/watch?v=XxFGokyYo6g&gl=US&hl=en Wayback Machine]{{cbignore}}: {{Cite web|last=Albin|first=Pierre|date=2019|title=History of algebraic topology|website=[[YouTube]] |url=https://www.youtube.com/watch?v=XxFGokyYo6g}}{{cbignore}}</ref>{{Rp|0:53:00}} Therefore, if X and Y are homeomorphic then they are homotopy-equivalent, but the opposite is not true. Some examples:

* A solid disk is homotopy-equivalent to a single point, since you can deform the disk along radial lines continuously to a single point. However, they are not homeomorphic, since there is no [[bijection]] between them (since one is an infinite set, while the other is finite).
* The [[Möbius strip]] and an untwisted (closed) strip are homotopy equivalent, since you can deform both strips continuously to a circle. But they are not homeomorphic.