  Hausdorff relation, topologically identifying points in a given space,
belongs to elementary tools of modern mathematics. We show that if subtle
enough mathematical methods are used to analyze this relation, the conclusions
may be far-reaching and illuminating. Examples of situations in which the
Hausdorff relation is of the total type, i.e., when it identifies all points of
the considered space, are the space of Penrose tilings and space-times of some
cosmological models with strong curvature singularities. With every Hausdorff
relation a groupoid can be associated, and a convolutive algebra defined on it
allows one to analyze the space that otherwise would remain intractable. The
regular representation of this algebra in a bundle of Hilbert spaces leads to a
von Neumann algebra of random operators. In this way, a probabilistic
description (in a generalized sense) naturally takes over when the concept of
point looses its meaning. In this situation counterparts of the position and
momentum operators can be defined, and they satisfy a commutation relation
which, in the suitable limiting case, reproduces the Heisenberg indeterminacy
relation. It should be emphasized that this is neither an additional assumption
nor an effect of a quantization process, but simply the consequence of a purely
geometric analysis.
