Metric and topological notions in

Bricogne, G.

doi:10.1107/97809553602060000551

International
Tables for
Crystallography
Volume B
Reciprocal space
Edited by U. Shmueli

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International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 26 | 1 | 2 |

Section 1.3.2.2.1. Metric and topological notions in $[{\bb R}^{n}]$

G. Bricogne^a

^a MRC Laboratory of Molecular Biology, Hills Road, Cambridge CB2 2QH, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.2.1. Metric and topological notions in $[{\bb R}^{n}]$

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The set $[{\bb R}^{n}]$ can be endowed with the structure of a vector space of dimension n over $[{\bb R}]$ , and can be made into a Euclidean space by treating its standard basis as an orthonormal basis and defining the Euclidean norm : $[\|{\bf x}\| = \left({\textstyle\sum\limits_{i = 1}^{n}} x_{i}^{2}\right)^{1/2}.]$

By misuse of notation, x will sometimes also designate the column vector of coordinates of $[{\bf x} \in {\bb R}^{n}]$ ; if these coordinates are referred to an orthonormal basis of $[{\bb R}^{n}]$ , then $[\|{\bf x}\| = ({\bf x}^{T} {\bf x})^{1/2},]$ where $[{\bf x}^{T}]$ denotes the transpose of x.

The distance between two points x and y defined by $[d({\bf x},{\bf y}) = \|{\bf x} - {\bf y}\|]$ allows the topological structure of $[{\bb R}]$ to be transferred to $[{\bb R}^{n}]$ , making it a metric space. The basic notions in a metric space are those of neighbourhoods, of open and closed sets, of limit, of continuity, and of convergence (see Section 1.3.2.2.6.1).

A subset S of $[{\bb R}^{n}]$ is bounded if sup $[\|{\bf x} - {\bf y}\| \;\lt\; \infty]$ as x and y run through S; it is closed if it contains the limits of all convergent sequences with elements in S. A subset K of $[{\bb R}^{n}]$ which is both bounded and closed has the property of being compact, i.e. that whenever K has been covered by a family of open sets, a finite subfamily can be found which suffices to cover K. Compactness is a very useful topological property for the purpose of proof, since it allows one to reduce the task of examining infinitely many local situations to that of examining only finitely many of them.

References

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 26