I've worked out that the book was right after all - though the reasoning is not obvious (to me). The claim is: Let

be a Hausdorff space and

is a locally compact subspace. If

then there is a closed neighbourhood

of

such that

is a compact neighbourhood of

in the subspace topology of

.

Proof of claim: As

is locally compact we have a compact n'hood

of

. This means that

is compact in both

and

and there is an open

containing

such that

. Now

is locally compact and Hausdorff and this means that it is a regular space. So since

is an open n'hood of

in the regular space

, there is an open n'hood

of

such that

. It is easy to check that

is our desired closed n'hood of

.

Phew.