The alphabet for this question is $=\{0,1\}.$ The operation SW is defined to be:

$SW(L)=\{0w0|winL\}$

where $L$ is a language over $.$

In other words, SW$($L$)$ is the result of "sandwiching" each word in $L$ between two O$\text{'}$s$.$

To prove that the class of regular languages is closed under SW$,$ we can use the following construction: Given a DFA $M=(Q,\{0,1\},,q0,F),$ we will build an NFA N such that $L(N)=SW(L(M)).$

$N=(Q\{A1,A2\},\{0,1\},{}^{\text{'}},A1,\{A2\})$

Such that ${}^{\text{'}}$ is defined below:

${}^{\text{'}}(A1,0)=\{q0\}$

${}^{\text{'}}(q,x)=\{(q,x)\}if$ qinQ,$q!$inF,xin$\{0,1\}$

${}^{\text{'}}(q,1)=\{(q,1)\}if$ qinF,

${}^{\text{'}}(q,0)=\{(q,0),A2\}if$ ainF