# $NL^2 = NDSPACE(\log^2n)$ is closed under complement

• From Savitch's theorem we have $$NL^2 \subseteq L^4$$, which is deterministic and thus closed under complement.
• From Immerman–Szelepcsényi theorem we have $$NL = coNL$$.

Why then $$NL^2 = coNL^2$$

The Immerman–Szelepcsényi theorem states that $$\mathsf{NSPACE}(s(n))$$ is closed under complementation whenever $$s(n) \geq \log n$$.