Let $K$ be a non-archimedean field. For $n \in \mathbb{N}$ let 

\begin{equation*}
T_n=\{ \sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in  K[\![X_1,\ldots, X_n ]\!] \mid a_{\nu_1, \ldots, \nu_n } \rightarrow 0 \text{ for } \sum \nu_i\rightarrow \infty \}
\end{equation*}

be the Tate algebra. For $f=\sum_{\nu_1, \ldots, \nu_n \geq 0 } a_{\nu_1, \ldots, \nu_n }X_1^{\nu_1}\cdots X_n^{\nu_n} \in T_n$ the Gauss norm $|\, \,\, |$ ist defined by  $|f|=\max_{(\nu_1, \ldots, \nu_n)} |a_{\nu_1, \ldots, \nu_n }|$. Then $T_n$ is complete with respect to $|\, \,\, |$ and $K[X_1,\ldots, X_n]$ is dense in it. 

For any (closed) ideal $\mathfrak{a} \subset T_n $, we have for $\overline{f} \in T_n/\mathfrak{a}$ the residue norm defined by $|\overline{f}|_{res}=\inf \{|h| \mid h \in \overline{f}\}$. Then $T_n/\mathfrak{a}$ complete with respect to the residue norm. 

Now let $I \subset K[X_1,\ldots, X_n]$ be closed with respect to the Gauß norm. Then the residue semi-norm on $K[X_1,\ldots, X_n]/I$ is actually a norm. 

Then I have the following question. 

 1. Are there Ideals $I \subset K[X_1,\ldots, X_n]$ which are non-closed? 
 2. Is the completion of $I$ with respect to the Gauss norm $I\cdot T_n$?
 3. Is the completion of $K[X_1,\ldots, X_n]/I$ with respect to the residue norm $T_n/(I\cdot T_n)$?