Details
| Original language | English |
|---|---|
| Pages (from-to) | 121-146 |
| Number of pages | 26 |
| Journal | Categories and General Algebraic Structures with Applications |
| Volume | 6 |
| Issue number | SpecialIssue 1 |
| Publication status | Published - Jan 2017 |
Abstract
We show that in ZF set theory without choice, the Ultrafilter Principle (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 UP but also from 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.
Keywords
- (super)compact, Choice, foot, free semilattice, locale, noetherian, prime, sober, well-filtered
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Discrete Mathematics and Combinatorics
- Mathematics(all)
- Computational Mathematics
- Mathematics(all)
- Applied Mathematics
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Categories and General Algebraic Structures with Applications, Vol. 6, No. SpecialIssue 1, 01.2017, p. 121-146.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - Choice principles and lift lemmas
AU - Erné, Marcel
PY - 2017/1
Y1 - 2017/1
N2 - We show that in ZF set theory without choice, the Ultrafilter Principle (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 UP but also from 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.
AB - We show that in ZF set theory without choice, the Ultrafilter Principle (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 UP but also from 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.
KW - (super)compact
KW - Choice
KW - foot
KW - free semilattice
KW - locale
KW - noetherian
KW - prime
KW - sober
KW - well-filtered
UR - http://www.scopus.com/inward/record.url?scp=85045237239&partnerID=8YFLogxK
M3 - Article
AN - SCOPUS:85045237239
VL - 6
SP - 121
EP - 146
JO - Categories and General Algebraic Structures with Applications
JF - Categories and General Algebraic Structures with Applications
SN - 2345-5853
IS - SpecialIssue 1
ER -