# open subspace of locally compact

It is kown that

1. A closed subspace of a locally compact space is locally compact
2. If $X$ is locally compact Hausdorff and a dense subspace $Y\subseteq X$ is locally compact iff $Y$ is open.

From the informations above, I have questions as follow:

1. If $X$ is locally compact, is there any open subspace $Y$ of $X$ which is not locally compact?

2. If $X$ is locally compact Hausdorff, is there any closed subspace $Y$ of $X$ which is not locally compact?

• Isn't question 2 answered by fact 1? And open subspaces of locally compact spaces are locally compact as well. More generally, if $Y$ is the intersection of an open and a closed subspace, then it is locally compact. – Stefan Hamcke Mar 3 '15 at 1:33