  In this paper, we introduce the notion of expanding topological space. We
define the topological expansion of a topological space via local
multi-homeomorphism over coproduct topology, and we prove that the coproduct
family associated to any fractal family of topological spaces is expanding. In
particular, we prove that the more a topological space expands, the finer the
topology of its indexed states is. Using multi-homeomorphisms over associated
coproduct topological spaces, we define a locally expandable topological space
and we prove that a locally expandable topological space has a topological
expansion. Specifically, we prove that the expanding fractal manifold is
locally expandable and has a natural topological expansion.
