Alexandrov one-point compactification

A special technique, the Alexandrov one-point compactification method, often used by topologists within a differential-topological framework, has been applied in the proof of the Holographic Electron Density Fragment Theorem [159-161],... 

Some non-compact topological spaces (X, T) can be converted into some compact topological spaces (Xx, Tx) by a technique called the Alexandrov one-point compactification. Here... 

More details of examples of the chemical applications of the Alexandrov one- point compactification method can be found in Refs. [118] and [159]. 

Based on the tools employed in the Hohenberg-Kohn theorem, also in part on the result of Riess and Miinch, and on a four-dimensional version of the Alexandrov one-point compactification method of topology applied to the complete three-dimensional electron density, it was possible to prove recently that for nondegenerate ground-state electron densities, the Holographic Electron Density Theorem applies any nonzero volume part of the nondegenerate ground-state electron density cloud contains all information about the molecule [4,5]. 

It will be shown in the next section that by using a four-dimensional electron density model and the Alexandrov one-point compactification of the ordinary three-dimensional space R, it will be possible to use analyticity arguments on compact sets to establish the claim that the electron density of any finite subsystem of nonzero volume determines the electron density of the rest of the system. 

Figure 1. An illustration of the Alexandrov one-point compactification, as applied to a two-dimensional plane. See text for details.

The three-dimensional molecular electron density p(r) fulfills all the continuity, differentiability, and exponential convergence conditions specified for the function F(x) of the two-dimensional example. Consequently, a four-dimensional variant of the Alexandrov one-point compactification is feasible, and the three-dimensional space can be replaced by a three-dimensional sphere 5 embedded in a four-dimensional space R, using a one-to-one assignment of points r of space to the points r of the 3-sphere S. In addition, a single point, a formal north pole n of the sphere corresponds to all formal points of infinite displacement from the center of mass of the molecule. 

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