On elementary equivalence and isomorphism of clone segments (Q1922122)

For topological spaces on the same underlying set there are three notions of `proximity' of their clone segments: equality, isomorphism, and elementary equivalence. Clearly, equality implies isomorphism, which implies elementary equivalence. The reverse of the above implications fails in the strongest way: More precisely, a principal application of the main theorem is the following: For any choice of \(1\leq n\leq m\leq p\) of integers two topological spaces \(X\) and \(Y\) are constructed on the same underlying set such that the initial \(k\)-segments of their clones of continuous maps coincide exactly when \(k\leq n\), are isomorphic exactly when \(k\leq m\), and are elementary equivalent exactly when \(k\leq p\).