Expansions which introduce no new open sets

Expansions which introduce no new open sets

We consider the question of when an expansion of a topological structure has the property that every open set definable in the expansion is definable in the original structure. This question is related to and inspired by recent work of Dolich, Miller and Steinhorn on the property of having o-minimal open core. We answer the question in a fairly general setting and provide conditions which in pr...

Semi-θ-Open Sets and New Classes of Maps

semi-θ-open sets and new classes of maps

Expansions of the real line by open sets: o-minimality and open cores

The open core of a structure R := (R, <, . . .) is defined to be the reduct (in the sense of definability) of R generated by all of its definable open sets. If the open core of R is o-minimal, then the topological closure of any definable set has finitely many connected components. We show that if every definable subset of R is finite or uncountable, or if R defines addition and multiplication ...

Expansions of the Real Line by Open Sets: O-minimality and Open Cores

The open core of a structure R := (R; <; : : :) is deened to be the reduct (in the sense of deenability) of R generated by all of its deenable open sets. If the open core of R is o-minimal, then the topological closure of any deenable set has nitely many connected components. We show that if every deenable subset of R is nite or uncountable, or if R deenes addition and multiplication and every ...