Closure Properties of Locally Testable Language
A language is $k$-locally testable if it is a Boolean combination of languages of the form $\{u\}$, $vA^*$, $A^*v$ or $A^*wA^*$, with $|u| &lt; k - 1$, $|v| = k - 1$ and $|w| = k$. Thus, yes, $k$-locally testable languages are closed under complementation.

But the link you gave to Wikipedia gives the definition of local languages, which is not the same thing as locally testable languages.
And indeed, local languages are not closed under complementation.

Finally, a local automaton recognizing a language $L$ might be incomplete. Therefore, you first need to complete it to compute the complement of $L$ by exchanging final and non final states. But this completed automaton is no longer local...