TY - JOUR
T1 - Generic left-separated spaces and calibers
AU - Juhász, I.
AU - Shelah, S.
PY - 2003/8/1
Y1 - 2003/8/1
N2 - In this paper we use a natural forcing to construct a left-separated topology on an arbitrary cardinal κ. The resulting left-separated space Xκ is also 0-dimensional T2, hereditarily Lindelöf, and countably tight. Moreover if κ is regular then d(Xκ)=κ, hence κ is not a caliber of Xκ, while all other uncountable regular cardinals are. This implies that some results of [A.V. Archangelskiicaron;, Topology Appl. 104(2000) 13-16] and [I. Juhász, Z. Szentmiklóssy, Topology Appl. 119 (2002) 315-324] are, consistently, sharp.We also prove it is consistent that for every countable set A of uncountable regular cardinals there is a hereditarily Lindelöf T3 space X such that Q=cf(Q)>ω is a caliber of X exactly if Q∉A.
AB - In this paper we use a natural forcing to construct a left-separated topology on an arbitrary cardinal κ. The resulting left-separated space Xκ is also 0-dimensional T2, hereditarily Lindelöf, and countably tight. Moreover if κ is regular then d(Xκ)=κ, hence κ is not a caliber of Xκ, while all other uncountable regular cardinals are. This implies that some results of [A.V. Archangelskiicaron;, Topology Appl. 104(2000) 13-16] and [I. Juhász, Z. Szentmiklóssy, Topology Appl. 119 (2002) 315-324] are, consistently, sharp.We also prove it is consistent that for every countable set A of uncountable regular cardinals there is a hereditarily Lindelöf T3 space X such that Q=cf(Q)>ω is a caliber of X exactly if Q∉A.
KW - Caliber
KW - Density
KW - Hereditarily Lindelöf
KW - Left separated
KW - Tightness
UR - http://www.scopus.com/inward/record.url?scp=30244459994&partnerID=8YFLogxK
U2 - 10.1016/S0166-8641(02)00367-X
DO - 10.1016/S0166-8641(02)00367-X
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AN - SCOPUS:30244459994
SN - 0166-8641
VL - 132
SP - 103
EP - 108
JO - Topology and its Applications
JF - Topology and its Applications
IS - 2
ER -