No. Let $X$ be a compactly generated Hausdorff space which fails to be locally compact at precisely one point. Now take the Alexandroff compactification of $X$, adding exactly one new point, whose neighborhoods have compact complement in $X$. The new space is not Hausdorff, but has the property that it is compact, and each compact subspace is closed. In general the image of a compact space is compact, and hence each retract of the new space is a closed subspace of itself.
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Modifed the original answer, added the extra assumption that $X$ is compactly generated.
Paul Fabel
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Paul Fabel
- 2k
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