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Difference between revisions of "Proper map"

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(Created page with "{{TEX|done}} {{MSC|54C05}} Let $X$ and $Y$ be two topological spaces. A continuous map $f:X\to Y$ is called ''proper'' if $f^{-1}...")
 
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When $X$ is [[Hausdorff space|Hausdorff]] and $Y$ [[Locally compact space|locally compact]] the properness of $f$ is equivalent to the requirement that $f^{-1} (\{y\})$ is compact for every $y\in Y$.
 
When $X$ is [[Hausdorff space|Hausdorff]] and $Y$ [[Locally compact space|locally compact]] the properness of $f$ is equivalent to the requirement that $f^{-1} (\{y\})$ is compact for every $y\in Y$.
  
If $X$ is a compact space and $Y$ is Hausdorff, then any proper map $f:X\to Y$ is [[Closed mapping|closed]].
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If $X$ is a compact space and $Y$ is Hausdorff, then any continuous map $f:X\to Y$ is proper and [[Closed mapping|closed]].

Latest revision as of 09:32, 14 January 2014

2020 Mathematics Subject Classification: Primary: 54C05 [MSN][ZBL]

Let $X$ and $Y$ be two topological spaces. A continuous map $f:X\to Y$ is called proper if $f^{-1} (K)$ is compact for every $K\subset Y$ compact.

When $X$ is Hausdorff and $Y$ locally compact the properness of $f$ is equivalent to the requirement that $f^{-1} (\{y\})$ is compact for every $y\in Y$.

If $X$ is a compact space and $Y$ is Hausdorff, then any continuous map $f:X\to Y$ is proper and closed.

How to Cite This Entry:
Proper map. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Proper_map&oldid=31244