# Three sides lemma

From Topospaces

*This term is nonstandard and is being used locally within the wiki. For its use outside the wiki, please define the term when using it.*

*This article is about the statement of a simple but indispensable lemma in topology*

## Statement

The inclusion of three sides in the unit square is equivalent to the inclusion of one side in the unit square, viz., is equivalent to the inclusion of a boundary in .

An explicit isotopy can also be written down, which is useful in some situations.

## Applications

The crux of the application of the three sides lemma is that it allows us to extend homotopies given both *initial* conditions and *final* conditions. The homotopy to be extended here is the middle side, and the initial and final conditions are the extreme sides.

- The three sides lemma is sometimes used to give a different definition of the second homotopy group of a pair. In fact, the generalized three sides lemma is used to give alternative definitions for all the higher homotopy groups.
- The three sides lemma gives a quick proof that the inclusion of the two endpoints in the unit interval is a cofibration (contrast this with the two sides lemma, which shows that the inclusion of one endpoint is a cofibration). An explicit retraction can also be written down.
- The three sides lemma gives a proof that the inclusion of the disjoint union of the center and boundary sphere of a disc, in the disc, is a cofibration. This is used to prove that the inclusion of a point in a manifold is a cofibration
`Further information: Center plus boundary in disc is cofibration` - The three sides lemma is also used in proofs involving fibrations (or more generally, Serre fibrations). For instance, it shows that for a Serre fibration, the fibers at two points in the same path component are homotopy-equivalent.
`Further information: Fibers are homotopy-equivalent for Serre fibrations`