open subspace of locally compact

Question

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?

Answer 1

The definition of locally compact can be: (i) every point has a ndhd basis of compact nbhds, or the weaker (ii) every point has a compact neighborhood. In case the space is Hausdorff, both are equivalent, and then, the locally compact subspaces are exactly the locally closed subsets, that is, the intersections of an open and a closed subset. Clearly, for the definition (i) open subsets are always locally compact. For (ii) however, there are compact (non-Hausdorff) spaces which have non locally compact open subsets. Besides the usual bibliography, one can always take a look at the Steen-Seebach counterexample bible!