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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?