If I have a topological space $X$ and a subspace $Y$ which is locally compact then does this mean that $X$ is also locally compact? For local compactness, I want to show that every point has a neighborhood base consisting of compact sets (I'm not assuming $X$ is Hausdorff).

Is it true that if $x\in X$ and $K$ is a compact neighbourhood of $x$ in $Y$ then is it also a neighbourhood of $x$ in $X$?

Many thanks!

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    $\begingroup$ Consider a subspace consisting of a single point. $\endgroup$ – David Mitra Dec 17 '14 at 12:40
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    $\begingroup$ Or take the disjoint union of a locally compact space and a space that isn't. $\endgroup$ – Ayman Hourieh Dec 17 '14 at 12:42
  • $\begingroup$ Or just $\emptyset \subset X$... $\endgroup$ – Najib Idrissi Dec 17 '14 at 12:54

As people in the comments have pointed out, no this isn't necessarily true. In my experience in topology, a trait $T$ of a subspace $Y$ usually tells you nothing about the space $X$. Most typically you must first know that $X$ has the trait $T$ before even considering if $Y$ has $T$. If $Y$ has $T$ we call $Y$ "Hereditarily $T$". It is also very common that $Y$ has to be closed and or compact if it is to be hereditarily $T$.


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