Choice principles and lift lemmas

We show that in {\bf ZF} set theory without choice, the Ultrafilter \mbox{Principle} ({\bf UP}) is equivalent to several compactness theorems for Alexandroff discrete spaces and to Rudin's Lemma, a basic tool in topology and the theory of quasi\-continuous domains. Important consequences of Rudin's Lemma are various lift lemmas, saying that certain properties of posets are inherited by the free unital semilattices over them. Some of these principles follow not only from {\bf UP} but also from {\bf DC}, the Principle of Dependent Choices. On the other hand, they imply the Axiom of Choice for countable families of finite sets,which is not provable in ZF
set theory.