Abstract
It is shown that for a compact Desarguesian projective Klingenberg plane P with incidence structure P=〈#x2119;, \(\mathbb{L}\), I〉 and neighbour relation ∼, where two distinct points always lie on some line, exactly one of the following holds: P is a non-discrete connected or totally disconnected ordinary projective plane with ∼=id, P is a finite projective plane with ∼=id, P is a finite projective Hjelmslev plane with ∼≠id, or P is a non-discrete totally disconnected ordinary projective plane with ∼≠id.
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