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    \begin{document}
    \begin{center}
    \textbf{ MATH534A, Problem Set 7, due October 30 }
    \end{center}
    All problems are worth the same number of points.
    \begin{enumerate}
    \item Let $F_i \colon M \to N_i$, where $i \in \{1, \dots, k\}$, be smooth maps, and assume that one of them is an immersion. Show that the map $F \colon M \to N_1 \times \dots \times N_k$, given by $F(x) = (F_1(x), \dots, F_k(x))$, is an immersion.
    \item \begin{enumerate}
    \item 
    Let $F \colon \R^n \to \R^n$ be a map whose components are given by \begin{align*}f_i = 
    \left[
    \begin{aligned}
    \,x_i \sqrt{1+\sum_{i = k+1}^nx_i^2} \quad\mbox{ for } i \leq k,\\
    \,x_i \phantom{ \sqrt{1+\sum_{i = k+1}^nx_i^2}} \quad\mbox{ for } i > k,
    \end{aligned}
    \right.
\end{align*}
where $k \in  \{1, \dots, n\}$ is given.
Show that $F$ is a diffeomorphism.
\item Show that the restriction of $F$ to a subset of $\R^n$ given by $\sum_{i = 1}^k x_i^2 = 1$ is an embedding.
\item Hence show that the subset of $\R^n$ given by
$$
\sum_{i = 1}^k x_i^2 \,\,- \sum_{i = k+1}^n x_i^2 = 1
$$
is a submanifold diffeomorphic to $S^{k-1} \times \R^{n - k}$. (By definiton, $S^0$ is a two-point set with discrete topology.)
    \end{enumerate}
    \item Let $M$ be a compact manifold, and let $C_1, C_2 \subset M$ be its closed disjoint subsets. Let also $f_1, f_2$ be smooth functions defined on some open neighborhoods of $C_1, C_2$ respectively. Show that there exists a smooth function $f \colon M \to \R$ such that $f\vert_{C_1} = f_1$, $f\vert_{C_2} = f_2$.
      \item Let $M$ be a compact manifold, $N$ be its compact submanifold, and let $f \colon N \to \R$ be a smooth function. Show that there exists a smooth function $\tilde f \colon M \to \R$ such that $\tilde f\vert_{N} = f$.
    \item Prove that any open cover of a second countable topological space admits a countable subcover.
    \end{enumerate}
    \end{document}