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Let $N$ be a unipotent algebraic group and $X,Y$ be two algebraic subsets of $N$. It is known that if $X,Y$ are algebraic subgroups of $N$, then the product $X\cdot Y$ is closed (algebraic subset).

My questions:

1). Is $X\cdot Y$ closed for every closed subsets $X,Y$ of $N$?

2). Is $X\cdot Y$ closed in the case when only one of $X,Y$ is an algebraic subgroup?

Improve this question
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    $\begingroup$ Both are false even in the additive group $\mathbf{C}^2$. Take $X$ the hyperbola $\{(x,y):xy=1\}$ and $Y$ the closed subgroup $\{(x,y):y=0\}$. Then $X+Y=\{(x,y):y\neq 0\}$ is not closed. $\endgroup$
    – YCor
    Commented Apr 2, 2021 at 6:04
  • $\begingroup$ Great answer! Thank you. $\endgroup$
    – Vladimir47
    Commented Apr 2, 2021 at 13:32

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