Let $ L = \{ \langle M \rangle : L(M) = \Sigma^* -\{\epsilon\} \}$
Determine whether $ L\in RE, \space L\in R, \space L\in co-RE$, or none of the above.
I tried to show a reduction function from $L_{\Sigma^*}$ to $L$ to prove that $L\notin RE$ , here is my attempt:
$\forall M\in L_{\Sigma^*} : f(M) = M'$, where M' is a Turing machine which works as follows:
$\forall w\in \Sigma^*: $ if $w=\epsilon$ accept w; Otherwise run M on w and answer as M.
I manged to show that if $X \in L_{\Sigma^*}$ than $f(X)\in L$, but the other direction ($X \notin L_{\Sigma^*}$ than $f(X)\notin L$) seems to be more problematic and I'm not sure that reduction will even work.
I will appreciate any direction.
Btw, I'm new to reduction so please try to be patient and understanding.
Thanks!