  In the framework of a recently proposed topological approach to phase
transitions, some sufficient conditions ensuring the presence of the
spontaneous breaking of a Z_2 symmetry and of a symmetry-breaking phase
transition are introduced and discussed. A very simple model, which we refer to
as the hypercubic model, is introduced and solved. The main purpose of this
model is that of illustrating the content of the sufficient conditions, but it
is interesting also in itself due to its simplicity. Then some mean-field
models already known in the literature are discussed in the light of the
sufficient conditions introduced here.
