  Factorization algebras have been defined using three different topologies on
the Ran space. We study these three different topologies on the exponential,
which is the union of the Ran space and the empty configuration, and show that
an exponential property is satisfied in each case. As a consequence we describe
the weak homotopy type of the exponential Exp(X) for each topology, in the case
where X is not connected.
  We also study these exponentials as stratified spaces and show that the
metric exponential is conically stratified for a general class of spaces. As a
corollary, we obtain that locally constant factorization algebras defined by
Beilinson-Drinfeld are equivalent to locally constant factorization algebras
defined by Lurie.
