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

International Tables for Crystallography (2010). Vol. B, ch. 1.3, pp. 27-28   | 1 | 2 |

Section 1.3.2.2.6.1. General topology

G. Bricognea

aGlobal Phasing Ltd, Sheraton House, Suites 14–16, Castle Park, Cambridge CB3 0AX, England, and LURE, Bâtiment 209D, Université Paris-Sud, 91405 Orsay, France

1.3.2.2.6.1. General topology

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Most topological notions are first encountered in the setting of metric spaces. A metric space E is a set equipped with a distance function d from [E \times E] to the non-negative reals which satisfies:[\matrix{(\hbox{i})\hfill & d(x, y) = d(y, x)\hfill &\forall x, y \in E\hfill &\hbox{(symmetry)\semi}\hfill\cr\cr (\hbox{ii})\hfill &d(x, y) = 0 \hfill &\hbox{iff } x = y\hfill &\hbox{(separation)\semi}\hfill\cr\cr (\hbox{iii})\hfill & d(x, z) \leq d(x, y) + d(y, z)\hfill &\forall x, y, z \in E\hfill &\hbox{(triangular}\hfill\cr& & &\hbox{inequality).}\hfill}]By means of d, the following notions can be defined: open balls, neighbourhoods; open and closed sets, interior and closure; convergence of sequences, continuity of mappings; Cauchy sequences and completeness; compactness; connectedness. They suffice for the investigation of a great number of questions in analysis and geometry (see e.g. Dieudonné, 1969[link]).

Many of these notions turn out to depend only on the properties of the collection [{\scr O}(E)] of open subsets of E: two distance functions leading to the same [{\scr O}(E)] lead to identical topological properties. An axiomatic reformulation of topological notions is thus possible: a topology in E is a collection [{\scr O}(E)] of subsets of E which satisfy suitable axioms and are deemed open irrespective of the way they are obtained. From the practical standpoint, however, a topology which can be obtained from a distance function (called a metrizable topology) has the very useful property that the notions of closure, limit and continuity may be defined by means of sequences. For nonmetrizable topologies, these notions are much more difficult to handle, requiring the use of `filters' instead of sequences.

In some spaces E, a topology may be most naturally defined by a family of pseudo-distances [(d_{\alpha})_{\alpha \in A}], where each [d_{\alpha}] satisfies (i) and (iii) but not (ii). Such spaces are called uniformizable. If for every pair [(x, y) \in E \times E] there exists [\alpha \in A] such that [d_{\alpha} (x, y) \neq 0], then the separation property can be recovered. If furthermore a countable subfamily of the [d_{\alpha}] suffices to define the topology of E, the latter can be shown to be metrizable, so that limiting processes in E may be studied by means of sequences.

References

Dieudonné, J. (1969). Foundations of Modern Analysis. New York, London: Academic Press.








































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