Locally compact subspaces are open in Hausdorff compactifications
= Locally compact subspaces are open in Hausdorff compactifications
If a locally <compact> <Hausdorff space> is densely embedded in a <Hausdorff space>, its image is <open>. Around a point choose an ambient <open set> whose trace lies in a <compact> <neighborhood> $K$. Density puts that ambient <open set> inside the ambient <closure> of its trace, which lies in $K$ because <compact subsets> of a <Hausdorff space> are <closed>. Thus the ambient <open set> lies in the image.