Decomposability and symmetric continua in [S^2]

In this note we discuss symmetric continua in the 2-sphere [S^2], i.e. continua invariant under the antipodal map. Our results indicate an essential difference between decomposable and indecomposable elements in this class. Symmetric decomposable continua can ahvays be written in a natural form [union of sets C and (-C)], where C is a non-symmetric subcontinuum of [S^2] connecting two antipodal points. Symmetric indecomposable continua are quite specific objects and can be determined by the remarkable property of being irreducible between each pair of its antipodal points. We also show that such spaces do exist, see Section 4 for a relevant construction of a symmetric indecomposable circle-like continuum.