On the one hand, the ideals of a well quasi-order (wqo) naturally form a compact topological space into which the wqo embeds. On the other hand, Nash-Williams' barriers are given a uniform structure by embedding them into the Cantor space. We prove that every map from a barrier into a wqo restricts on a barrier to a uniformly continuous map, and therefore extends to a continuous map from a countable closed subset of the Cantor space into the space of ideals of the wqo. We then prove that, by shrinking further, any such continuous map admits a canonical form with regard to the points whose image is not isolated. cr As a consequence, we obtain a simple proof of a result on better quasi-orders (bqo); namely, a wqo whose set of non-principal ideals is a bqo is actually a bqo.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-3-2,
author = {Rapha\"el Carroy and Yann Pequignot},
title = {From well to better, the space of ideals},
journal = {Fundamenta Mathematicae},
volume = {227},
year = {2014},
pages = {247-270},
zbl = {06354992},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-3-2}
}
Raphaël Carroy; Yann Pequignot. From well to better, the space of ideals. Fundamenta Mathematicae, Tome 227 (2014) pp. 247-270. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm227-3-2/