We extend the cactus theorem of Dinitz, Karzanov, Lomonosov to metric spaces. In particular we show that if is a separable continuum which is not separated by points then the set of all –tuples of points separating can be encoded by an –tree.
"The cactus tree of a metric space." Algebr. Geom. Topol. 11 (5) 2547 - 2578, 2011. https://doi.org/10.2140/agt.2011.11.2547