A cute and thought-provoking theorem for sure, but arguably none of the three given criteria for clustering are well motivated, so the result is much less relevant than usually claimed.

- scale-invariance: stretching data along some dimensions should not change clustering.

This is clearly not true: . . . (three well-spaced spots) may be reasonably seen as three clusters, whereas ||| (three nearby elongated bars) not.

- richness: all groupings must be reachable.

Also not quite true, both of the two cases: (1) all clusters are singleton points and (2) a single cluster that contains all points, mean the same: no useful cluster structure found. So it is enough if one of these groupings are reachable, and not both.

- consistency: increasing inter-cluster differences and decreasing intra-cluster differences should not change clustering.

Also not quite true: suppose we have 9 clusters:

  . . .
  . . .
  . . .

now move the points so that the columns get further apart, at some point we will get: | | |, where 3 clusters are more reasonable.