On the Compatibility of Metric and Linear Order

Abstract

There is a classic question of introducing some reasonable order on the real plane R2.

We consider this problem through the prism of the relation between metric proximity and linear order proximity, introducing a way to determine the compatibility relation between metric and linearly ordered spaces, and study the basic properties of this connection.

For R and its metric subspaces, the order on the line is compatible with the metric according to
the definition we introduce. It turns out that for Rn for n > 1, as well as for other similar metric spaces in which there is a subspace isometric to an open ball in Rn, it is impossible to introduce an order that is compatible with the metric.

At the same time, we also introduce a family of spaces called discrete curved lines, which are not generally isometric to the subspaces of R, but for which it is possible to introduce a linear order compatible with the natural metric on them. For them we prove sufficient conditions for the compatibility of the metric with the order. Using them, we construct spaces with non-trivial
properties in which the order is compatible with the metric. In particular, we show that although on all of Rn no order is compatible with the metric, in any normed space one can introduce non-trivial metric subspaces compatible with some order (in the case of infinite-dimensional normed spaces, this subspace can be constructed to be non-embeddable in any finite-dimensional subspace).