  We prove that if an RCD space has a regular isometric immersion in a
Euclidean space, then the immersion is a locally bi-Lipschitz embedding map.
This result leads us to prove that if a compact non-collapsed RCD space has an
isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is
a locally bi-Lipschitz embedding map to a sphere, which generalizes a
fundamental theorem of Takahashi in submanifold theory to a non-smooth setting.
Applications of these results include a topological sphere theorem and
topological finiteness theorems, which are new even for closed Riemannian
manifolds.
