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A Quasicontinuous Domain Without an Interval Retract
Abstract
We give negative answers to two questions of X.Xu concerning the occurrence of the unit interval in nonquasialgebraic quasicontinuous domains. We construct, directly from the binary tree and the binary-value map, a well-founded quasicontinuous dcpo P=2<ωmathbin∪̇[0,1] which is not quasialgebraic. Moreover, P contains no sub-dcpo isomorphic to [0,1], and [0,1] is not a Scott-continuous retract of P.
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BibTeX
@misc{shen2026quasicontinuous,
title = {{A Quasicontinuous Domain Without an Interval Retract}},
author = {Chong Shen and Xiaoyong Xi and Dongsheng Zhao},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20284},
eprint = {2609.20284},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20284},
}
RIS
TY - UNPB
TI - A Quasicontinuous Domain Without an Interval Retract
AU - Chong Shen
AU - Xiaoyong Xi
AU - Dongsheng Zhao
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.20284
UR - https://arxiv.org/abs/2609.20284
AB - We give negative answers to two questions of X.Xu concerning the occurrence of the unit interval in nonquasialgebraic quasicontinuous domains. We construct, directly from the binary tree and the binary-value map, a well-founded quasicontinuous dcpo P=2^<ωmathbin∪̇[0,1] which is not quasialgebraic. Moreover, P contains no sub-dcpo isomorphic to [0,1], and [0,1] is not a Scott-continuous retract of P.
ER -