Question

*7. (Stack of Records Theorem.) Suppose that y is a regular value of f: X$\to$ Y, where X is compact and has the same dimension as Y. Show that $f^{-1}(y)$ is a finite set {$x_1,...,x_N$}. Prove there exists a neighbor- hood U of y in Y such that $f^{-1}(U)$ is a disjoint union $V_1 \cup \dots \cup V_N$, where $V_i$ is an open neighborhood of $x_i$, and f maps each $V_i$ diffeo- morphically onto U. [HINT: Pick disjoint neighborhoods $W_i$ of $x_i$ that are mapped diffeomorphically. Show that f(X$\setminus \cup W_i$) is compact and does not contain y.) See Figure 1-13.

          *7. (Stack of Records Theorem.) Suppose that y is a regular value of
f: X$\to$ Y, where X is compact and has the same dimension as Y. Show
that $f^{-1}(y)$ is a finite set {$x_1,...,x_N$}. Prove there exists a neighbor-
hood U of y in Y such that $f^{-1}(U)$ is a disjoint union $V_1 \cup \dots \cup V_N$,
where $V_i$ is an open neighborhood of $x_i$, and f maps each $V_i$ diffeo-
morphically onto U. [HINT: Pick disjoint neighborhoods $W_i$ of $x_i$ that
are mapped diffeomorphically. Show that f(X$\setminus \cup W_i$) is compact and
does not contain y.) See Figure 1-13.

*7. (Stack of Records Theorem.) Suppose that y is a regular value of
f: X→ Y, where X is compact and has the same dimension as Y. Show
that f^-1(y) is a finite set x1,...,xN. Prove there exists a neighbor-
hood U of y in Y such that f^-1(U) is a disjoint union V1 ∪…∪ VN,
where Vi is an open neighborhood of xi, and f maps each Vi diffeo-
morphically onto U. [HINT: Pick disjoint neighborhoods Wi of xi that
are mapped diffeomorphically. Show that f(X∖∪ Wi) is compact and
does not contain y.) See Figure 1-13.

Added by Jeff B.

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