Can closed dense sets in the completion space be no longer closed?

Question

Given a metric space $(E,d)$, where $d$ is the metric, let $T$ refer to the topology in $E$ induced by $d$. Suppose that given $T$, $E$ is closed.

Let $F$ refer to the completion of $E$ with respect to $d$, i.e. $F$ can be formed from the equivalence classes of Cauchy sequences of $E$, and $(F,d)$ is now the complete metric space with topology $T'$ induced by $d$.

Let $\psi:E \rightarrow F$, be a morphism such that $\psi$ embeds any $x \in E$ to its 'equivalent' element $\psi(x) \in F$, i.e. $\psi(x) = \{x,x,x,x,...\} \in F$

I would like to ask, even if $E$ is closed under $T$, $\psi(E)$ is no longer closed under $T'$ right ?

Answer 1

Consider the usual Q and its completion R.
Let p be an irrational point of R.
I = (p - 1, p + 1) is clopen within Q.
Within R, I is open and not closed.