quotient map of locally compact spaces

Question

Suppose $X,Y$ are non-compact locally comapct Hausdorff spaces,suppose we have a quotient map $Q:X\to Y$,can we conclude that $Q$ is proper?

Answer 1

No.

Let $X=\mathbb{R}^2$ and $Y=\mathbb{R}$.

Let $Q(x,y)=x$.

It is not proper as $Q^{-1}(0)\cong\mathbb{R}$.