Lyusternik-Shnirel'man-Borsuk covering theorem

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A theorem usually stated as follows:

1) Each closed covering of contains at least one set with .

Contrary to the equivalent Borsuk–Ulam theorem, it seems to be not common to use the same name also for the following equivalent symmetric versions:

2) Let be closed sets with . If , then .

3) [a2].

In all these results, the estimates are optimal (in 3), in fact, equality holds). It is worth mentioning that 2) gave the motivation for the notion of the genus of a set symmetric with respect to a free -action.

For other equivalent versions and for generalizations to coverings involving other symmetries (e.g. with respect to free -actions), cf. [a3] and the references therein.

One major field of applications are estimates of the number of critical points of even functionals; this can be used, e.g., in the theory of differential equations.


[a1] K. Borsuk, "Drei Sätze über die —dimensionale Sphäre" Fund. Math. , 20 (1933) pp. 177–190
[a2] L. Lyusternik, L. Shnirel'man, "Topological methods in variational problems" , Issl. Inst. Mat. Mekh. OMGU (1930) (In Russian)
[a3] H. Steinlein, "Spheres and symmetry: Borsuk's antipodal theorem" Topol. Methods Nonlinear Anal. , 1 (1993) pp. 15–33
How to Cite This Entry:
Lyusternik-Shnirel'man-Borsuk covering theorem. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by H. Steinlein (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article