Let $M$ be a manifold – by which I mean a second countable Hausdorff smooth manifold.
Here's an "obviously" correct definition of (embedded) submanifold:
Definition A. A subspace $S$ of $M$ is a submanifold if there is an atlas $\{ \phi_\alpha : U_\alpha \to U'_\alpha \}$ for $S$ such that each $\phi^{-1}_\alpha : U'_\alpha \to U_\alpha$ is a smooth immersion (or, equivalently, embedding) considered as a map to $M$.
But what if we relax these conditions slightly?
Definition B. A subspace $S$ of $M$ is a submanifold if there is some smooth structure on $S$ making the inclusion $S \hookrightarrow M$ into a smooth immersion.
This definition makes it clear that the image of any smooth embedding is automatically a submanifold, but it is not clear how the smooth structure of $S$ is induced from $M$, nor whether such a smooth structure is unique. A little thought shows that definitions A and B are equivalent.
Definition C. A subspace $S$ of $M$ is a submanifold if it has the following property: for all points $p$ in $S$, there exist an open neighbourhood $U$ of $p$ in $M$ and a chart $\phi : U \to U'$ such that $S \cap U$ is the inverse image of a linear subspace of $U'$.
This definition leaves no room for doubt about the existence and uniqueness of a smooth structure of $S$ and gives a clear, concrete picture of how it is induced from the smooth structure of $M$. But on the other hand, one worries that this definition is too restrictive: it seems to imply that every submanifold can be locally thickened without self-intersection, which sounds like a non-trivial condition to me.
Question. Are definitions A, B, and C equivalent? If they are, a proof-sketch or reference would be much appreciated.
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Best Answer
For compact manifolds with boundary and $S$ a neat submanifold, or for $S$ closed, this is Theorem II.2.3 in Kosinski's Differential Manifolds. It is indeed a nontrivial fact, and it indeed implies that a neat submanifold has a collar. If $S$ is allowed to have corners, it needs also to be "regular" for the definitions to coincide- the reference is Proposition A.3 in Farber's Topology of Closed One-Forms.