2025/05/26 by Basset, Estelle, Lancien, Gilles, Procházka, Antonín · 1 citation
#30L05 #46B20 #46B22 #51F30 #54E45 #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2505.19891
We show that there are uncountably many mutually non-isomorphic Lipschitz-free spaces over countable, complete, discrete metric spaces. Also there is a countable, complete, discrete metric space whose free space does not embed into the free space of any uniformly discrete metric space. This enhanced diversity is a consequence of the fact that the dentability index D presents a highly non-binary behavior when assigned to the free spaces of metric spaces outside of the oppressive confines of compact purely 1-unrectifiable spaces. Indeed, the cardinality of \D(\mathcal F(M)): M countable, complete, discrete\ is uncountable while \D(\mathcal F(M)):M infinite, compact, purely 1-unrectifiable\=\ω,ω2\. Similar barrier is observed for uniformly discrete metric spaces as higher values of the dentability index are excluded for their free spaces: \D(\mathcal F(M)):M infinite, uniformly discrete\=\ω2,ω3\.