Normal spanning trees exist in a connectoid precisely when its ground set admits a well-ordering of countable separation number, via an extension of Jung's dispersed-sets characterization.
Pitz,Proof of Halin’s normal spanning tree conjecture, Israel Journal of Mathematics246 (2021), no
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Connectoids II: existence of normal trees
Normal spanning trees exist in a connectoid precisely when its ground set admits a well-ordering of countable separation number, via an extension of Jung's dispersed-sets characterization.