Help in understanding product topology

Question

Let $X_i$ be topological space. Let $O_i \subseteq X_i$ denote open set and let $C_i \subseteq X_i$ denote closed set. Let $O$ be open set in $X = \prod_i X_i$ and let $\pi_i : X \to X_i$ be projection map.

Then $O = \bigcap_{i=0}^n \pi_i^{-1}O_i$ for some $O_i$ open in $X_i$. Let $C$ be closed in $X$. I am wondering how to imagine $C$? I know $C = \bigcup_{i \in I} \bigcap_{k=0}^{n_i} \pi_k^{-1}C_k$ for $C_k$ closed in $X_k$?

What about $\pi_i C$? Does it hold that $\pi_iC = C_i$ for $C_i$ closed in $X_i$? Thank you

Answer 1

It may be helpful for you, however I'm not sure.

The example which is mentioned by Brian Rushton shows that the projections are not closed:

The projection $p: \mathbb R^2 \rightarrow \mathbb R$ of the plane $ \mathbb R^2 $ onto the $x$-axis is not closed. Indeed, the set $\color{red}{F}=\{(x,y)\in \mathbb R^2 : xy=1\}$ is closed in $\mathbb R^2 $ and yet its image $\color{blue}{p(F)}= \mathbb R \setminus \{0\}$ is not closed in $\mathbb R$.

Answer 2

Not all open sets $O$ are equal to the intersection of finitely many $O_i$; only the basis elements are.

Also, $\pi_i$ does not send closed sets to closed sets; look at the hyperbola $xy=1$ in $\mathbb{R}^2$.