Quotient of a locally compact space

Question

I am looking for an example of a quotient of a locally compact space that isn't locally compact.

Is there a not too complicated example ?

Answer 1

$X=\mathbb{R}$, and identify the integers to a point. The resulting quotient map is an example.

In this answer you can see the proof of the fact that the quotient is not locally compact, and it is also noted that this quotient map is even closed.