  In the usual brane-world scenario matter fields are confined to the
four-dimensional spacetime, called a 3-brane, embedded in a higher-dimensional
space, usually referred to as the bulk spacetime. In this paper we assume that
the 3-brane is a de Sitter space; there is only one extra spatial dimension,
assumed to be time dependent. By using the form of the brane-world
energy-momentum tensor suggested by Shiromizu et al. in the five-dimensional
Einstein equations, it is proposed that the cosmological expansion of the
3-brane may provide a possible explanation for the collapse of the extra
dimension, as well as for the energy stored in the resulting curled-up
dimension. More precisely, whenever the bulk cosmological constant \Lambda is
negative, the extra spatial dimension rapidly shrinks during the inflation of
the brane. When \Lambda is positive, on the other hand, the extra spatial
dimension either completely follows the cosmological expansion of the brane or
completely ignores it, thereby shrinking relative to the expanding space. This
behavior resembles the all-or-nothing behavior of ordinary systems in an
expanding universe, as recently demonstrated by R.H. Price.
