Quotient map with finite fibre, locally compact source and non-locally compact target?

Question

What is an example of a quotient map $f\colon X \to Y$ such that $f$ have finite fibres, $X$ be locally compact and $Y$ be not locally compact?

Answer 1

Here's a simple example. Let $A=(0,1]\times[0,1]$ and $B=[0,1]$, let $X=A\coprod B$, and let $f:X\to Y$ be the quotient that identifies $(t,0)\in A$ with $t\in B$ for each $t\in(0,1]$. Then $Y$ is not locally compact at $f(0)$, since any neighborhood will contain a sequence of the form $f(s_n,t_n)$ where $s_n\to 0$ and $t_n>0$ and such a sequence has no accumulation point.