# Local Parametrizations of Immersed Submanifold - John Lee's Smooth Manifolds, p. 111

The following is the definition that I'm confused about:

Suppose $$S\subseteq M$$ is an immersed $$k$$-dimensonal submanifold. A local parametrization of S is a continuous map $$X: U\to M$$ whose domain is an open subset $$U\subseteq \mathbb{R}^k$$, whose image is an open subset of $$S$$, and which, considered as a map into $$S$$, is a homeomorphism onto its image.

By "considered as a map into $$S$$," does Dr. Lee mean that $$X(U)$$ is homeomorphic to a subset of $$S$$, where $$S$$ has its submanifold topology or the subspace topology with respect to $$M$$ (since $$X$$ has codomain $$M$$)?

It means that if you consider the map $$X: U \rightarrow X(U)$$, this is a homeomorphism if you give to the set $$X(U)$$ the subspace topology inherited by $$S$$. While, $$S$$ has the topological structure given by the immersion, since it is an immersed submanifold.