Difference between revisions of "Proper map"
From Encyclopedia of Mathematics
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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 | + | 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
Proper map. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Proper_map&oldid=31244