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I want to find a relation between proper map and continuous closed map $f$ where $f$ is a mapping of metric space $X$ onto a space $Y$.

Whether this is true or can be found an counterexample?

If $f$ is a continuous closed mapping of a metrizable space $X$ onto a space $Y$ then for every $y\in$ $Y$, $f^{-1}(y)$ is compact ?

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    @IsaacSolomon: I recently became familiar with this site I am not familiar with all its features. Thanks for your note.2012-11-26

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Let $X$ be any noncompact metric space and $Y=\{y\}$ a one point space. Then the constant map $f: X \rightarrow Y$ is continuous and closed, but $f^{-1}(y)$ is not compact.