ИСТИНА

The class of all metric spaces considered up to an isometry and endowed with the Gromov--Hausdorff distance is investigated. This distance is equal (up to multiplication by $2$) to the least possible distortion of metric over all correspondences (multivalued analogues of bijections) between those spaces. If one restricts himself by compact metric spaces, then the Gromov--Hausdorff distance is a metric. The resulting metric space $\cM$ is referred as the Gromov--Hausdorff space, and many its properties are well-studied. In particular, this space is path-connected, Polish (i.e., separable and complete), and geodesic, see details in~\cite{BurBurIva}. Here we investigate the Gromov--Hausdorff distance on the class $\GH$ of all metric spaces considered up to an isometry. To work with such a ``monster--space'' we use the von Neumann--Bernays--G\"odel axioms system that permits to define correctly a distance function even on such a proper class. We suggest a way to define a topology on $\GH$ by means of a filtration over cardinality. As a result, we define continuous curves in $\GH$ and show that the Gromov--Hausdorff distance is an intrinsic extended pseudometric. We also investigate the geometry of metric segments in $\GH$, where a metric segment is defined as a class of points lying between two fixed ones. We show that a metric segment could be a proper class, not a set (Conjecture: It is always the case). Besides, we study the possibility to extend a metric segment (to another one) beyond some its endpoint. This problem turns out to be rather non-trivial, and its complete solution is unknown even for the Gromov--Hausdorff space $\cM$. The key result gives some sufficient condition of non-extendability of a metric segment beyond one of its endpoints. We also give several interesting examples.