Antichains in the Wadge Order for Connected Spaces

For any non-totally disconnected Polish space, there is a family of c = 2א0 many Wadge incomparable finite level Borel subsets. If the space is additionally locally compact or locally connected, there is a family of 2 many Wadge incomparable subsets. In this note, we study the Wadge order for Polish spaces which are not totally disconnected. For a fixed Polish space, the Wadge order for the space compares its subsets via continuous functions; for subsets A and B, A is Wadge reducible to B or A ≤ B if there is a continuous function from the space to itself under which A is the preimage of B. Thus a preorder is associated to every Polish space. The structure of the Wadge order for the Borel subsets of Baire space ω is well understood and characterized by two important facts, the first of which is the semilinear ordering principle or Wadge’s lemma; for any two Borel subsets A and B of Baire space, A ≤ B or B ≤ ω−A. The second is the wellfoundedness of the Wadge order restricted to the Borel sets, proved by Martin and Monk. These facts are true for all subsets of Baire space under the axiom of determinacy and the same proofs generalize. They further hold for all zero dimensional Polish spaces. For Polish spaces which are not zero dimensional, the structure of the Wadge order may be more complicated. For example, Wadge’s lemma fails for the real line as noted in [5, remark 9.26] (see also [2, example 3]). The counterexample [2] shows that there are three sets of reals which are pairwise incomparable in the Wadge order. In this paper we provide conditions which imply that the Wadge order of a space has large antichains, i.e. collections of incomparable sets. For any Polish space which is not totally disconnected, we construct a family of c = 2א0 many incomparable finite level Borel subsets. In particular, the semilinear ordering principle fails for these spaces (but the proof of this alone is much shorter). As an application of the proof technique, we give a sufficient condition on the space for ill-foundedness of the Wadge order.