How to show that the set of TMs that accept languages of size two is recognizable?
Based on RanG. answer in comments.

To show ${Lx}$ is unrecognizable, show $\overline{Htm}$ &lt;= ${Lx}$

Given $\overline{Htm}$ = {M description: M is a TM and M loops on ''}

def R(&lt;M&gt;):
    def N(x):
        if x == 0 or x == 1 then accept
        M('')
        return accept
    return &lt;N&gt;



  If M is in $\overline{Htm}$ then M loops then N will accept either 0 or 1 then
  |L(N)| = 2 then N is in ${Lx}$ 
  
  if M is not in $\overline{Htm}$ then M halts then N
  will accept all x so |L(N)| = $\infty$ then N is not in ${Lx}$