  We show that every possible metric associated with critical ($\gamma=2$)
Liouville quantum gravity (LQG) induces the same topology on the plane as the
Euclidean metric. More precisely, we show that the optimal modulus of
continuity of the critical LQG metric with respect to the Euclidean metric is a
power of $1/\log(1/|\cdot|)$. Our result applies to every possible
subsequential limit of critical Liouville first passage percolation, a natural
approximation scheme for the LQG metric which was recently shown to be tight.
