Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The paper is aimed at studying the topological dimension for sets definable in akly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the akly o-minimal context.  prove that the topological dimension of a set definable in a akly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [MMS] for sets and functions definable in models of akly o-minimal theories.  pay special attention to large subsets of Cartesian products of definable sets, showing that if X,Y and S are non-empty definable sets and S is a large subset of X ◊Y , then for a large set of tuples ha1,...,a2ki 2 X 2 k , where k = dim(Y ), the union of fibers Sa1 [... [Sa 2k is large in Y . Finally, given a akly o-minimal structure M,  find various conditions equivalent to the fact that the topological dimension in M enjoys the addition property.
['Roman Wencel']
I am writing a research paper on a idea Topological properties of sets definable in weakly o-minimal structures
[{"from": "human", "value": "I am writing a research paper on a idea Topological properties of sets definable in weakly o-minimal structures"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The paper is aimed at studying the topological dimension for sets definable in akly o-minimal structures in order to prepare background for further investigation of groups, group actions and fields definable in the akly o-minimal context.  prove that the topological dimension of a set definable in a akly o-minimal structure is invariant under definable injective maps, strengthening an analogous result from [MMS] for sets and functions definable in models of akly o-minimal theories.  pay special attention to large subsets of Cartesian products of definable sets, showing that if X,Y and S are non-empty definable sets and S is a large subset of X \u25caY , then for a large set of tuples ha1,...,a2ki 2 X 2 k , where k = dim(Y ), the union of fibers Sa1 [... [Sa 2k is large in Y . Finally, given a akly o-minimal structure M,  find various conditions equivalent to the fact that the topological dimension in M enjoys the addition property."}]