# 0708-1300/not homeomorphic

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Assume Failed to parse (lexing error): \~{f} : R^n --> R^m

```is a homeomorphism. Since Failed to parse (lexing error): \~{f}
is proper
```

we can extend it to a continuous map $f : S^n --> S^m$ which in fact will be a homeomorphism. Taking inverse if necessary we may assume $n < m$. Let Failed to parse (lexing error): F : Sn × [0, 1] --> Sm

```be a homotopy of $f$ to a smooth map i.e. $F$ is continuous,
```

$F(x, 0) = f(x)$ and $F(x, 1)$ is smooth. Since $F(x, 1)$ is smooth and $n < m$ all of its image points are singular values and by Sard's theorem constitute a set of measure zero. Then there is a point in $S^m$ not in the image of $F(x, 1)$, but the complement of that point is contractible. This means that we can extend $F$ to Failed to parse (lexing error): \~{F} : Sn × [0, 2] --> S^m

```to be a homotopy of $f$ to a constant map. But then
```

Failed to parse (lexing error): f^{－1}\circ\~{F}

```is a contraction of $S^n$ which is a contradiction with the fact that no
```

such contraction exists.