  We prove a general self-improvement property for a family of weighted
pointwise inequalities on open sets, including pointwise Hardy inequalities
with distance weights. For this purpose we introduce and study the classes of
$p$-Poincar\'e and $p$-Hardy weights for an open set $\Omega\subset X$, where
$X$ is a metric measure space. We also apply the self-improvement of weighted
pointwise Hardy inequalities in connection with usual integral versions of
Hardy inequalities.
