This is a very elementary question I'm a little confused about. 

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> Can any subset of $\mathbb{R}^2$ be expressed in form $A\times B$, where $A$ and $B$ are subsets of $\mathbb{R}$? 
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I'm thinking that it might not necessarily be so. For instance, if we consider the intervals $[0,2]$ and $[3,5]$ in $\mathbb{R}$, and throw out a single point, say $(1,4)$, out of $[0,2]\times [3,5]$ then there do not exist subsets $A$ and $B$ of $\mathbb{R}$ such that $A\times B = ([0,2]\times [3,5]) \setminus \{(1,4)\}$. Because if they were equal, surely $A$ would be equal to $[0,2]$ and $B$ would be equal to $[3,5]$. But as there is no way to put the constraint that $1$ cannot pair up with $4$, $A\times B$ will always be $[0,2]\times [3,5]$.

Am I thinking in the right direction? Thanks.