Example of $X$ such that $\pi_Y:X\times Y\to Y$ is not a closed map.

Question

I am reading Topology and Geometry by Bredon, and I just read this Proposition:

Proposition 8.2: If $X$ is compact then the projection $\pi_Y:X\times Y\to Y$ is closed.

Now, I do see how compactness is used in the proof of this statement, but it isn't obvious to me why this statement does not hold in general. Could anyone provide me with an example for which this breaks down if $X$ is not compact?

Answer 1

Take $X=Y=\mathbb{R}$ and let $A=\big\{(n, \frac{1}{n})\big\}_{n=1}^{\infty}$. The set $A$ is closed (these points are isolated from each other) but $\pi_Y(A)=\{\frac{1}{n}\}_{n=1}^{\infty}$ is not.