How to show that the set of TMs that accept languages of size two is recognizable? I know how to show $\overline{Lx}$ is unrecognizable.
I know how to show Lx is undecidable.

I would like the mapping reduction function that shows that Lx is recognizable or unrecognizable.

For instance, to show $\overline{Lx}$ is unrecognizable, show $\overline{Htm}$ &lt;= $\overline{Lx}$

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

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



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


I would like a similar proof to show that Lx is either recognizable or unrecognizable.
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 ${Lx}$