Countable discrete extensions of compact lines

Abstract

We consider a separable compact line K and its extension L consisting of K and countably many isolated points. The main object of study is the existence of a bounded extension operator E : C(K) → 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 greater than or equal to the least cardinality of a set X ⊆ [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.