Let $X$ be a locally compact Hausdorff space, $Y$ a Hausdorff space and
$f : X \to Y$
a Borel map. Is $f(Y)$ locally compact?
Images of locally compact spaces through Borel maps
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general-topology
measure-theory
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0ok. it seems that the answer is "no": even continuity seems to be not enough, see http://math.stackexchange.com/questions/1287344/continuous-image-of-a-locally-compact-space-is-locally-compact?rq=1 – 2017-01-09
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1no, every space is the continuous image of a discrete space , which are locally compact. – 2017-01-09