Three sides lemma

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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.


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 I in I \times I.

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


  • 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