Example of a surjective local homeomorphism that is not a covering?

Question

Let $X$ and $Y$ be connected, locally path connected, and Hausdorff topological spaces. Can someone give me an example of a surjective local homeomorphism that is not a covering?

I don't think this is the same question as here? There's different conditions?

Answer 1

Restrict the usual covering projection from the real line to the circle(that is $x$ goes to $e^{2\pi ix}$) on the positive side of the line.

$p:R_+\rightarrow S^1$ defined as $p(x)=(\cos2\pi x,\sin2\pi x)$

This is a local homeomorphism but not covering projection. As the point $(1,0)$ on the circle is not evenly covered by the projection map.