TY - JOUR

T1 - A Q-WADGE HIERARCHY IN QUASI-POLISH SPACES

AU - Selivanov, Victor

PY - 2022/6/5

Y1 - 2022/6/5

N2 - The Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff-Kuratowski (HK)-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

AB - The Wadge hierarchy was originally defined and studied only in the Baire space (and some other zero-dimensional spaces). Here we extend the Wadge hierarchy of Borel sets to arbitrary topological spaces by providing a set-theoretic definition of all its levels. We show that our extension behaves well in second countable spaces and especially in quasi-Polish spaces. In particular, all levels are preserved by continuous open surjections between second countable spaces which implies e.g., several Hausdorff-Kuratowski (HK)-type theorems in quasi-Polish spaces. In fact, many results hold not only for the Wadge hierarchy of sets but also for its extension to Borel functions from a space to a countable better quasiorder Q.

KW - better quasiorder

KW - Borel hierarchy

KW - fine hierarchy

KW - h-quasiorder

KW - iterated labeled tree

KW - Q-partition

KW - Wadge hierarchy

UR - http://www.scopus.com/inward/record.url?scp=85132762602&partnerID=8YFLogxK

U2 - 10.1017/jsl.2020.52

DO - 10.1017/jsl.2020.52

M3 - Article

AN - SCOPUS:85132762602

VL - 87

SP - 732

EP - 757

JO - Journal of Symbolic Logic

JF - Journal of Symbolic Logic

SN - 0022-4812

IS - 2

ER -