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SEPARATING MANY LOCALISATION CARDINALS ON THE GENERALISED BAIRE SPACE
- Part of
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- Journal:
- The Journal of Symbolic Logic / Volume 89 / Issue 3 / September 2024
- Published online by Cambridge University Press:
- 13 April 2023, pp. 1212-1231
- Print publication:
- September 2024
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- Article
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Given a cofinal cardinal function
$h\in {}^{\kappa }\kappa $ for 
$\kappa $ inaccessible, we consider the dominating h-localisation number, that is, the least cardinality of a dominating set of h-slaloms such that every 
$\kappa $-real is localised by a slalom in the dominating set. It was proved in [3] that the dominating localisation numbers can be consistently different for two functions h (the identity function and the power function). We will construct a 
$\kappa ^+$-sized family of functions h and their corresponding localisation numbers, and use a 
${\leq }\kappa $-supported product of a cofinality-preserving forcing to prove that any simultaneous assignment of these localisation numbers to cardinals above 
$\kappa $ is consistent. This answers an open question from [3].
