  We consider a separable compact line $K$ and its extension $L$ consisting of
$K$ and a countable number of isolated points. The main object of study is the
existence of a bounded extension operator $E: C(K)\to C(L)$. We show that if
such an operator exists then there is one for which $\|E\|$ is an odd natural
number. We prove that if the topological weight of $K$ is bigger than or equal
to the least cardinality of a set $X \subseteq [0,1]$ that cannot be covered by
a sequence of closed sets of measure zero then there is an extension $L$ of $K$
admitting no bounded extension operator.
