General topology

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. 28 | 1 | 2 |

Section 1.3.2.2.6.1. General topology

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

1.3.2.2.6.1. 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: $[\matrix{(\hbox{i})\hfill & d(x, y) = d(y, x)\hfill &\forall x, y \in E\hfill &\hbox{(symmetry);}\hfill\cr\cr (\hbox{ii})\hfill &d(x, y) = 0 \hfill &\hbox{iff } x = y\hfill &\hbox{(separation);}\hfill\cr\cr (\hbox{iii})\hfill & d(x, z) \leq d(x, y) + d(y, z)\hfill &\forall x, y, z \in E\hfill &\hbox{(triangular}\hfill\cr& & &\hbox{inequality).}\hfill}]$ 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).

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 non-metrizable 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 and London: Academic Press.Google Scholar

International Tables for Crystallography (2006). Vol. B. ch. 1.3, p. 28