Let $M\subset \mathbb{R}^n$ be a $k$-dimensional manifold and $X\subset M$ a subset. The boundary of $X$ in $M$, denoted by $\partial_M X$, is the set of all $x\in X$ such that each neighborhoud of $x$ contains points in $X$ and $M\setminus X$.
I want to show that the `smooth boundary' of $X$, as defined here: Definition 6.6.2, is a smooth $(k-1)$-dimensional manifold.
By implicit function theorem I can see that in a small neighborhood $V$ of $x\in \partial_M X$, the locus $Y\subset V$ defined by $f=0, g=0$ is a $(k-1)$-dimensional manifold. But I can't see why each such point in the locus has to be a boundary point of $X$ with respect to $M$.
Any thoughts would be appreciated.
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