TY - JOUR
ID - 40448
TI - Choice principles and lift lemmas
JO - Categories and General Algebraic Structures with Applications
JA - CGASA
LA - en
SN - 2345-5853
AU - Ern\'e, Marcel
AD - Faculty for Mathematics and Physics, IAZD, Leibniz Universit\"at, Welfengarten 1, D 30167 Hannover, Germany.
Y1 - 2017
PY - 2017
VL - 6
IS - Speical Issue on the Occasion of Banaschewski's 90th Birthday (I)
SP - 121
EP - 146
KW - Choice
KW - (super)compact
KW - foot
KW - free semilattice
KW - locale
KW - noetherian
KW - prime
KW - sober
KW - well-filtered
DO -
N2 - We show that in ${\bf ZF}$ set theory without choice, the Ultrafilter 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 quasicontinuous 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 ${\bf ZF}$ set theory.
UR - https://cgasa.sbu.ac.ir/article_40448.html
L1 - https://cgasa.sbu.ac.ir/article_40448_f354e76a770fa82f66fa30955e1aba56.pdf
ER -