  For time-fractional parabolic equations with a Caputo time derivative of
order $\alpha\in(0,1)$, we give pointwise-in-time a posteriori error bounds in
the spatial $L_2$ and $L_\infty$ norms. Hence, an adaptive mesh construction
algorithm is applied for the L1 method, which yields optimal convergence rates
$2-\alpha$ in the presence of solution singularities.
