Details
| Original language | English |
|---|---|
| Pages (from-to) | 263-290 |
| Number of pages | 28 |
| Journal | Topology and its applications |
| Volume | 241 |
| Publication status | Published - 1 Jun 2018 |
Abstract
We investigate in ZF set theory without choice principles a general lattice-theoretical prime separation lemma and compare it with diverse statements about variants of the sobriety concept for topological spaces. Some of these properties coincide in the presence of choice principles but differ in their absence. UP, the Ultrafilter Principle (or, equivalently, the Prime Ideal Theorem) is not only equivalent to the Separation Lemma, but also necessary and sufficient for the desired coincidences. Furthermore, we prove the equivalence of UP to several statements about filtered systems of compact sets, among them the Hofmann–Mislove Theorems, several compact intersection theorems, and an irreducible transversal theorem. Moreover, many fundamental dualities between certain categories of topological spaces and categories of ordered structures turn out to be equivalent to UP. But we also give choice-free proofs for such dualities, amending slightly the involved definitions.
Keywords
- (Strictly) continuous, (Strictly) sober, (Strictly) spatial, Compact, Irreducible, Prime, Supercompact, Transversal, Well-filtered
ASJC Scopus subject areas
- Mathematics(all)
- Geometry and Topology
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In: Topology and its applications, Vol. 241, 01.06.2018, p. 263-290.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The strength of prime separation, sobriety, and compactness theorems
AU - Erné, Marcel
PY - 2018/6/1
Y1 - 2018/6/1
N2 - We investigate in ZF set theory without choice principles a general lattice-theoretical prime separation lemma and compare it with diverse statements about variants of the sobriety concept for topological spaces. Some of these properties coincide in the presence of choice principles but differ in their absence. UP, the Ultrafilter Principle (or, equivalently, the Prime Ideal Theorem) is not only equivalent to the Separation Lemma, but also necessary and sufficient for the desired coincidences. Furthermore, we prove the equivalence of UP to several statements about filtered systems of compact sets, among them the Hofmann–Mislove Theorems, several compact intersection theorems, and an irreducible transversal theorem. Moreover, many fundamental dualities between certain categories of topological spaces and categories of ordered structures turn out to be equivalent to UP. But we also give choice-free proofs for such dualities, amending slightly the involved definitions.
AB - We investigate in ZF set theory without choice principles a general lattice-theoretical prime separation lemma and compare it with diverse statements about variants of the sobriety concept for topological spaces. Some of these properties coincide in the presence of choice principles but differ in their absence. UP, the Ultrafilter Principle (or, equivalently, the Prime Ideal Theorem) is not only equivalent to the Separation Lemma, but also necessary and sufficient for the desired coincidences. Furthermore, we prove the equivalence of UP to several statements about filtered systems of compact sets, among them the Hofmann–Mislove Theorems, several compact intersection theorems, and an irreducible transversal theorem. Moreover, many fundamental dualities between certain categories of topological spaces and categories of ordered structures turn out to be equivalent to UP. But we also give choice-free proofs for such dualities, amending slightly the involved definitions.
KW - (Strictly) continuous
KW - (Strictly) sober
KW - (Strictly) spatial
KW - Compact
KW - Irreducible
KW - Prime
KW - Supercompact
KW - Transversal
KW - Well-filtered
UR - http://www.scopus.com/inward/record.url?scp=85045249772&partnerID=8YFLogxK
U2 - 10.1016/j.topol.2018.04.002
DO - 10.1016/j.topol.2018.04.002
M3 - Article
AN - SCOPUS:85045249772
VL - 241
SP - 263
EP - 290
JO - Topology and its applications
JF - Topology and its applications
SN - 0166-8641
ER -