  We define continuous C*-algebras over a topological space X and establish
some basic results. If X is a locally compact Hausdorff space, continuous
C*-algebras over X are equivalent to ordinary continuous C_0(X)-algebras. The
main purpose of our study is to prove that every continuous, full, separable,
nuclear C*-algebra over X is KK(X)-equivalent to a stable Kirchberg algebra
over X. (Here a Kirchberg algebra over X is a separable, nuclear, and strongly
purely infinite C*-algebra over X with primitive ideal space homeomorphic to
X.) In the case that X is a one-point space, this result is known as that every
separable nuclear C*-algebra is KK-equivalent to a stable Kirchberg algebra.
Moreover, as an intermediate result, we obtain the X-equivariant exact
embedding result for continuous C*-algebras over X.
