Jones Lemma

The lemma of Jones is a result from the mathematical sub-region of the topology which is based on the US mathematician F. Burton Jones back (1910-1999). It provides a criterion with which it can be shown that a topological space is not a normal space . The question of the normality of a topological space is significant because of the connection with the metrization problem , because a metric space is always normal.

Formulation of the result

Given a topological space and embedded in it two subspaces and , for which the following constraints are met: ${\displaystyle X}$ ${\displaystyle D_{0}\subseteq X}$ ${\displaystyle D_{1}\subseteq X}$

• ${\displaystyle D_{0}}$be a closed subspace of and with respect to the subspace topology discrete . ${\displaystyle X}$
• ${\displaystyle D_{1}}$lie close in . ${\displaystyle X}$
• Be it . ${\displaystyle |D_{0}|\geq 2^{|D_{1}|}}$

Then is not normal. ${\displaystyle X}$

Example: the Niemytzki room

The Niemytzki space , i.e. the closed upper half-plane , provided with the Niemytzki topology, fulfills the requirements of Jones' lemma with and . ${\displaystyle \Gamma }$ ${\displaystyle {\overline {\mathbb {H} }}\subset {\mathbb {R} \times \mathbb {R} }}$ ${\displaystyle D_{0}=\mathbb {R} \times \{0\}}$ ${\displaystyle D_{1}=({\mathbb {Q} \times \mathbb {Q} })\cap {\overline {\mathbb {H} }}}$

literature

items

• F. Burton Jones: Remarks on the Normal Moore Space Metrization Problem. In: RH Bing , Ralph J. Bean (Eds.): Topology Seminar Wisconsin, 1965 (= Annals of Mathematics Studies. Vol. 60, ISSN  0066-2313 ). Princeton University Press, Princeton NJ 1966, pp. 115-119.

Monographs

• James Dugundji : Topology . 8th printing. Allyn and Bacon, Boston MA 1973.
• Lutz Führer : General topology with applications . Vieweg, Braunschweig 1977, ISBN 3-528-03059-3 .
• Gregory Naber: Set-theoretic Topology. With Emphasis on Problems from the Theory of Coverings, Zero Dimensionality and Cardinal Invariants . University Microfilms International, Ann Arbor MI 1977, ISBN 0-8357-0257-X .
• Jun-iti Nagata : Modern General Topology (=  North Holland Mathematical Library . Volume 33 ). 2nd revised edition. North-Holland Publishing, Amsterdam / New York / Oxford 1985, ISBN 0-444-87655-3 ( MR0831659 ).
• Horst Schubert : Topology. An introduction . 4th edition. BG Teubner, Stuttgart 1975, ISBN 3-519-12200-6 .
• Stephen Willard: General Topology . Addison-Wesley, Reading MA et al. a. 1970.

Individual evidence

1. ^ Jones: Remarks on the Normal Moore Space Metrization Problem. In: Bing, Bean (Ed.): Topology Seminar Wisconsin, 1965. 1966, pp. 115–119, here p. 117.
2. ↑ Dugundji: Topology. 1973, p. 144.
3. ^ Willard: General Topology. 1970, p. 100.
4. ↑ Dugundji: Topology. 1973, p. 193.
5. ↑ Guide: General Topology with Applications. 1977, p. 127 ff.
6. ↑ Nagata: Modern General Topology. 1985, p. 244 ff.
7. ^ Schubert: Topology. 1975, p. 95 ff.
8. ^ Willard: General Topology. 1970, p. 161.
9. ^ Schubert: Topology. 1975, p. 78.
10. ^ Naber: Set-theoretic Topology. 1977, pp. 109-110.
11. ↑ Nagata: Modern General Topology. 1985, pp. 83-84.