Local Connectedness and Distance Functions

Abstract

Local connectedness functions for (κ, 1)-simplified morasses, localisations of the coupling function c studied in [M96, §1], are defined and their elementary properties discussed. Several different useful canonical ways of arriving at the functions are examined. This analysis is then used to give explicit formulae for generalisations of the local distance functions introduced with a recursive definition in [K00], leading to simple proofs of the principal properties of those functions. It is then also extended to the properties of local connectedness functions in the context of \( \kappa {\text{ - }}\mathbb{M} \)-proper forcing for successor κ. The functions are shown to enjoy substantial strengthenings of the properties (particularly the Δ-properties) hitherto proved for both the function c and for Todorcevic’s ρ-functions in the special case κ = ω1. A couple of examples of the use of local connectedness functions in consort with \( \kappa {\text{ - }}\mathbb{M} \)-proper forcing are then given.KeywordsDistance FunctionMaximal ElementCoupling FunctionSide ConditionLocal ConnectednessThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.