This is question 4 on page 92 in Chapter 16 of James R. Munkers’ Topology 2nd Edition textbook.

Question

A map \(f: X \rightarrow Y\) is said to be an open map if for every open set \(U\) of \(X\), the set \(f(U)\) is open in Y. Show that $\pi_1 : X \times Y \rightarrow X$ and $\pi_2 : X \times Y \rightarrow Y$ are open maps.

\(X\) and \(Y\) are both topologies.

Proof

For \(\pi_1 : X \times Y \rightarrow X\):

Let \(U\) be an open set of \(X\). Because \(\pi_1\) maps to the entirety of \(X\), discarding \(Y\) in the process, it follows that \(U\) must be in the resulting topological space \(X\).

Similar logic holds for \(\pi_2 : X \times Y \rightarrow Y\):

Let \(V\) be an open set of \(Y\). Because \(\pi_2\) maps to the entirety of \(Y\), discarding \(X\) in the process, it follows that \(V\) must be in the resulting topological space \(Y\).

\[\tag*{$\blacksquare$}\]

Extras

link to the mathjax LaTeX specification: https://treeofmath.github.io/tex-commands-in-mathjax/TeXSyntax.htm