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Two non-empty subsets $\mathsf{X}, \mathsf{Y}\subset \mathbb{R}^n$. Which conditions on $\mathsf{X}$ and $\mathsf{Y}$ are necessary and sufficient for $$ \mathsf{Z} = \mathsf{X}\times\mathsf{Y} $$ to be an open and connected subset of $\mathbb{R}^{2n}$?

Verónica Rmz.
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Euler_Salter
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  • According to this question, the product of two open sets is an open set.

    According to this question, the product of two connected sets is a connected set. Therefore if $\mathsf{X}$ and $\mathsf{Y}$ are open and connected, then so is $\mathsf{Z}$. These are sufficient conditions. What are necessary conditions though?

     – Euler_Salter Apr 21 '22 at 20:22
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    I think that the projections onto the two components from $\mathbb{R}^{2n}$ to $\mathbb{R}^n$ are open mappings. Moreover they are continuous. Thus they map connected sets to connected ones. – Jens Schwaiger Apr 21 '22 at 20:42

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