Let $(X_i)_{i \in \mathbb{N}},(f_{i,j})_{i\leq j \in \mathbb{N}})$ be an inverse system in $Top$ with inclusion $\iota_i:U \hookrightarrow X_i$ for all $i$, such that $f_{i-1,i}\circ\iota_i=\iota_{i-1}$. Therefore, we consider $U$ as a subspace of $X_i$. Let $X:=\varprojlim_{i \in \mathbb{N}}X_i$.
I found in the comments of Inverse Limit of Dense Subsets is Dense that if $U$ is dense in $X_i$ then it is dense in $X$, but I don't see why.
1st EDIT : My confusion was solved by the first comment. Nevertheless, my first question is
(1) Is $U$ dense in $X$ if $U$ is dense in $X_i$ for all $i$?
At next I am interested under which circumstance the converse holds.
(2) So if $U$ is dense in $X$, when is $U$ dense in $X_i$ for all $i$?
2nd EDIT: We can assume that each $X_i$ is a $K$-Banach space, i.e. $X$ is a Fréchet-space and $U$ is a subvector space. Not sure if this changes something.