(i) State the definition of smooth manifold with boundary and define the notion of boundary. Show that the boundary $\partial X$ is a manifold (without boundary) with $\operatorname{dim} \partial X=\operatorname{dim} X-1$.

(ii) Let $0<a<1$ and let $x_{1}, x_{2}, x_{3}, x_{4}$ denote Euclidean coordinates on $\mathbb{R}^{4}$. Show that the set

$X=\left\{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}-x_{4}^{2} \leqslant a\right\} \cap\left\{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=1\right\} \cap\left\{x_{1}^{2}+2 x_{2}^{2}+x_{3}^{2}+x_{4}^{2}=3 / 2\right\}$

is a manifold with boundary and compute its dimension. You may appeal to standard results concerning regular values of smooth functions.

(iii) Determine if the following statements are true or false, giving reasons:

a. If $X$ and $Y$ are manifolds, $f: X \rightarrow Y$ smooth and $Z \subset Y$ a submanifold of codimension $r$ such that $f$ is not transversal to $Z$, then $f^{-1}(Z)$ is not a submanifold of codimension $r$ in $X$.

b. If $X$ and $Y$ are manifolds and $f: X \rightarrow Y$ is smooth, then the set of regular values of $f$ is open in $Y$.

c. If $X$ and $Y$ are manifolds and $f: X \rightarrow Y$ is smooth then the set of critical points is of measure 0 in $X$.