Title: Isomorphism and classification for countable structures,
Abstract: We introduce a topology on the space of all isomorphism types represented in
a given class of countable models, and use this topology as an aid in
classifying the isomorphism types. This mixes ideas from effective descriptive
set theory and computable structure theory, extending concepts from the latter
beyond computable structures to examine the isomorphism problem on arbitrary
countable structures. We give examples using specific classes of fields and of
trees, illustrating how the new concepts can yield classifications that reveal
differences between seemingly similar classes. Finally, we use a computable
homeomorphism to define a measure on the space of isomorphism types of
algebraic fields, and examine the prevalence of relative computable
categoricity under this measure.