TY - JOUR
T1 - Cardinalities of topologies with small base
AU - Shelah, Saharon
PY - 1994/6/9
Y1 - 1994/6/9
N2 - Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T0). We prove that if T has a base of cardinality ≤μ, λ≤μ<2λ, λ strong limit of cofinality א0, then T has cardinality ≤μ or ≥2λ. This is our main conclusion (21). In Theorem 2 we prove it under some set-theoretic assumption, which is clear when λ = μ; then we eliminate the assumption by a theorem on pcf from [Sh 460] motivated originally by this. Next we prove that the simplest examples are the basic ones; they occur in every example (for λ = א0 this fulfills a promise from [Sh 454]). The main result for the case λ = א0 was proved in [Sh 454].
AB - Let T be the family of open subsets of a topological space (not necessarily Hausdorff or even T0). We prove that if T has a base of cardinality ≤μ, λ≤μ<2λ, λ strong limit of cofinality א0, then T has cardinality ≤μ or ≥2λ. This is our main conclusion (21). In Theorem 2 we prove it under some set-theoretic assumption, which is clear when λ = μ; then we eliminate the assumption by a theorem on pcf from [Sh 460] motivated originally by this. Next we prove that the simplest examples are the basic ones; they occur in every example (for λ = א0 this fulfills a promise from [Sh 454]). The main result for the case λ = א0 was proved in [Sh 454].
UR - http://www.scopus.com/inward/record.url?scp=38149145894&partnerID=8YFLogxK
U2 - 10.1016/0168-0072(94)90049-3
DO - 10.1016/0168-0072(94)90049-3
M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???
AN - SCOPUS:38149145894
SN - 0168-0072
VL - 68
SP - 95
EP - 113
JO - Annals of Pure and Applied Logic
JF - Annals of Pure and Applied Logic
IS - 1
ER -