  We study functions of least gradient as well as related superminimizers and
solutions of obstacle problems in metric spaces that are equipped with a
doubling measure and support a Poincar\'e inequality. We show a standard weak
Harnack inequality and use it to prove semicontinuity properties of such
functions. We also study some properties of the fine topology in the case
$p=1$. Then we combine these theories to prove a weak Cartan property of
superminimizers in the case $p=1$, as well as a strong version at points of
nonzero capacity. Finally we employ the weak Cartan property to show that any
topology that makes the upper representative $u^{\vee}$ of every
$1$-superminimizer $u$ upper semicontinuous in open sets is stronger (in some
cases, strictly) than the $1$-fine topology.
