ELESSAR TELKONT
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Homework Statement
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 X,Y be metric spaces and X\times Y with another metric the product metric space. A\subseteq X, B\subseteq Y be open sets, then A\times B\subseteq X\times Y is open.
Homework Equations
Definition of open and metric properties.
The Attempt at a Solution
The problem is possible to be reduced to the case where A,B are open balls in X,Y 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.