Example of a map which is not closed but every closed ball maps into closed set

Question

I'm trying to find a counter-example for the following statement: "Consider $X$ and $Y$ be any two metric spaces. If $f: X\rightarrow Y $ is a map such that $f$ maps closed balls in $X$ to closed Sets in $Y$ then $f$ is a closed map " I thought about inclusion maps , projection maps, constant maps, but I was unable to think counter example from there. Please help me on how to think of any counter example? I am a beginner in topology and metric spaces.

Answer 1

You can take, for instance, $\arctan\colon\Bbb R\longrightarrow\Bbb R$, with $\Bbb R$ endowed with the usual topology. It is not a closed map, since $\arctan(\Bbb R)=\left(-\frac\pi2,\frac\pi2\right)$. But it maps closed balls into closed sets.

Answer 2

why are you saying that projection maps would not work? What about the projection of the plane onto the real line? The graph of the hyperbola $h(x)=1/x$ is closed in the plane, but its projection is the real line minus the origin, which is not closed.