Let $f:(X,d)\to (X,d)$ be a bijection on metric space $(X,d)$. For $\epsilon>0$ the bijection map $f$ is called $\epsilon$_self_isometry, whenever
$$\sup_{x , x' \in X} \left| d( x,x') - d( f(x), f(x')) \right| \leq \epsilon $$
It is clear that if $f:(X,d) \to (X,d)$ is an isometry (that is $d(x,x')=d(g(x), g(x'))$), then it is an $\epsilon$_self_isometry, for all $\epsilon>0$ .
let $\Pi(X) = X \xrightarrow[onto]{1:1} X$. I defined the metric between $f,g \in \Pi(X)$ as
$$d_{\Pi(X)}(f,g) = \max_{x\in X} d(f(x),g(x)) = d(f(X),g(X)). $$
Question. Let $\epsilon>0$ be given.is dense sets $\textit{Iso}(X,d)$ , $\textit{Iso}_{\epsilon} (X,d)$ in $\Pi(X)$.