# What is the smooth structure of an embedded submanifold?

Here is a quotation from Chapter 5 in “Introduction to smooth manifolds” by J. Lee.

Suppose $$M$$ is a smooth manifold with or without boundary. An embedded submanifold of $$M$$ is a subset $$S subseteq M$$ that is a manifold (without boundary) in the subspace, endowed with a smooth structure with resect to which the inclusion map $$Sto M$$ is a smooth embedding.

My questions are the following:

• What is the topology on $$S$$?
• What is the smooth structure on $$S$$?

In my thought, $${i^{-1}(Ucap S)mid U in mathcal{O}(M)}$$ is the topology on $$S$$ with the topology $$mathcal{O}(M)$$ on $$M$$.

For the second question, I think $$mathcal{A}_{S}:={(i^{-1}(Ucap S), phicirc i)mid (U,phi)in mathcal{A}_{M}}$$ is the smooth structure on $$S$$ with the smooth structure $$mathcal{A}_{M}$$ on $$M$$, but it seems to be odd for me if $$mathrm{dim}(S) because the dimension of the image under $$phicirc i$$ is $$mathrm{dim}(M)$$, which is absurd.

Is there anything wrong with the above discussion?