Topologizing interpretable sets in O-minimal Structures

Let M be a structure in some language. Assume M has elimination of imaginaries. Let X be a definable set. Definable will mean “definable with parameters.” By a definable topology, we mean a definable family of subsets {By ⊂ X}y∈Y which form the basis for some topology on X. The fact that these form a basis for a topology amounts to the claim that if y1, y2 have By1 ∩By2 6= ∅, then for every x ∈ By1 ∩By2 there is a y3 ∈ Y such that x ∈ By3 ⊂ By1 ∩By2 . This is a first-order condition, so a definable topology on X remains a definable topology in elementary extensions of M . But note that if M M ′, the topology on X(M) need not agree with the subspace topology on X(M) as a subset of X(M ′). If D is a definable subset of X, and X has a definable topology, then the subspace topology on D is also definable. If X and Y are two sets with definable topologies, then the product topology on X × Y is definable. If X and Y are two sets with definable topologies, and f : X → Y is a definable function, then we can express whether or not f is continuous using some first-order statement. So the continuity of f is invariant under elementary extensions, and definable in families. We say that X is definably connected if there is no definable clopen set D with ∅ ( D ( X. The definable connectedness of X is invariant under elementary extensions, and typedefinable in families. If f : X → Y is a continuous definable function, and X is definably connected, then so is Y . A continuous map f : X → Y between abstract topological spaces is an open map if f(U) is open for every open subset U ⊂ X. If B is a basis of opens on X, it suffices to check U ∈ B. If f : X → Y is a continuous definable map between two definable topological spaces, then we can express that f is an open map via a first-order statement. So elementary extensions preserve whether or not f is open, and this is definable in families.