# Pitfalls of open maps

Suppose you have an open map $$p$$ between topological spaces, and if you have a subet $$A$$ of $$p$$’s domain such that $$p(A)$$ is open. Can you then conclude that $$A$$ is open? Nope! Consider the following spaces $$X=\{x_1,x_2\}$$ and $$Y=\{y_1,y_2\}$$ with topologies $$\tau_X=\{\varnothing, X, \{x_1\}\}$$ and $$\tau_Y=\{\varnothing,Y,\{y_1\}\}$$, respectively and let $$p: X\times Y\to X$$ be the projection onto its first fator. This is an open map. If we consider $$A=X\times\{y_2\}$$ we see that $$A$$ is not open in $$X\times Y$$, but we have that $$p(A)=p(X\times\{y_2\})= X$$ which is trivially open in $$X$$.