I found this question in an exercise on a book:

> 
> Check that any open or closed subset of a locally compact space is also locally compact.
> 
> 
> 

The definition of locally compact in the book is the following: a Hausdorff space is locally compact if for each point there is a compact neighborhood.

There is no other mention to this concept in the entire book, so I assume that a subset is locally compact if seeing it as a topological subspace then it is locally compact (Im not sure if this is the intended use of the concept for subsets).

But then is trivial that any subset of a locally compact space is also locally compact because any closed set and any open set of a subspace is induced by closed and open sets of the general space.

In other words: I cant see any reason that make me think that open or closed subspaces have a different behavior than any other subspace as locally compact spaces. It is my reasoning correct?

In other words: its possible that a subspace of a locally compact space would not be locally compact?