1. The problem statement, all variables and given/known data This is not really coursework. Instead, this is some sort of curiosity and proposition formulation rush. Then the initial questions are that if this is a valid result that is worth to be proven. Let [itex]X,Y[/itex] be metric spaces and [itex]X\times Y[/itex] with another metric the product metric space. [itex]A\subseteq X, B\subseteq Y[/itex] be open sets, then [itex]A\times B\subseteq X\times Y[/itex] is open. 2. Relevant equations Definition of open and metric properties. 3. The attempt at a solution The problem is possible to be reduced to the case where [itex]A,B[/itex] are open balls in [itex]X,Y[/itex] respectively, since one definition of openess is based on the fact that a every point in an open set has some neighborhood contained into the set. Then I think there must be a relationship between metrics on factor spaces and product space one. There is where I'm stuck.