# Relation between proper map and continuous closed map in metric space

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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Mr. Sina, it seems that you are not in the habit of accepting answers. This discourages people from trying to help you. Please try to accept answers in future (including accepting answers on past questions). –  Isaac Solomon Nov 26 '12 at 6:49
@IsaacSolomon: I recently became familiar with this site I am not familiar with all its features. Thanks for your note. –  M.Sina Nov 26 '12 at 8:02

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.