Give an example to show that if T is the product topology on X , then $\prod_{B}$ is need not be closed( Wayne patty 2.3.8)

Question

This question is from page85 of Wayne patty's topology.

Let {$(X_\alpha ,T_\alpha): \alpha \in \Lambda$} be an indexed family of topological spaces, let $X= \prod_{\alpha \in \Lambda}X_\alpha $ , let T be a topology on X, and for each $\beta \in \Lambda$, let $\pi_{\beta} : X \to X_{\beta}$ be the projective map. Then, Give an example to show that if T is the product topology on X, then $\pi_\beta$ need not be closed.

I am unable to construct such an example and so requesting you to help me.

Answer 1

The projection onto the first coordinate on $\mathbb R^{2}$ is not a closed map: $\{(a,b):ab=1\}$ is closed but its projection is not. The projection is $\mathbb R \setminus \{0\}$.