Closed embeddings into Lipscomb's universal space

Ivan Ivanšić (Author), Uroš Milutinović (Author)

Abstract

Naj bo ▫${\mathcal{J}}(\tau)$▫ Lipscombov enodimenzionalni prostor in ▫$L_n(\tau) = \{x \in {\mathcal{J}}(\tau)^{n+1}|$▫ vsaj ena koordinata od ▫{\sl x}▫ je iracionalna ▫$\} \subseteq {\mathcal{J}}(\tau)^{n+1}$▫ Lipscombov ▫$n$▫-dimenzionalni univerzalni prostor s težo ▫$\tau \ge \aleph_0$▫. V tem članku dokazujemo, da če je ▫$X$▫ poln metrizabilni prostor in velja ▫$\dim X \le n$▫, ▫$wX \le \tau$▫, tedaj obstaja zaprta vložitev prostora ▫$X$▫ v ▫$L_n(\tau)$▫. Še več, vsako zvezno funkcijo ▫$f: X \to {\mathcal{J}}(\tau)^{n+1}$▫ lahko poljubno natančno aproksimiramo z zaprto vložitvijo ▫$\psi: X \to L_n(\tau)$▫. Razen tega sta dokazani relativna verzija in punktirana verzija. V primeru separabilnosti je dokazan analogni rezultat, v katerem je klasična trikotna krivulja Sierpińskega (ki je homeomorfna ▫${\mathcal{J}}(3)$▫) nadomestila ▫${\mathcal{J}(\aleph_0)}$▫.

Keywords

matematika;topologija;dimenzija pokrivanja;posplošena krivulja Sierpińskega;univerzalni prostor;Lipscombov univerzalni prostor;vložitev;razširitev;poln metrični prostor;zaprta vložitev;mathematics;topology;covering dimension;embedding;closed embedding;generalized Sierpiński curve;universal space;Lipscomb universal space;complete metric space;extension;

Data

Language:	English
Year of publishing:	2006
Typology:	0 - Not set
Organization:	UM PEF - Faculty of Education
UDC:	515.127
COBISS:	14083417
ISSN:	1318-4865
Views:	760
Downloads:	76
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Unknown
Secondary title:	Zaprte vložitve v Lipscombov univerzalni prostor
Secondary abstract:	Let ▫${\mathcal{J}}(\tau)$▫ be Lipscomb's one-dimensional space and ▫$L_n(\tau) = \{x \in {\mathcal{J}}(\tau)^{n+1}\|$▫ at least one coordinate of ▫{\sl x}▫ is irrational ▫$\} \subseteq {\mathcal{J}}(\tau)^{n+1}$▫ Lipscomb's ▫$n$▫-dimensional universal space of weight ▫$\tau \ge \aleph_0$▫ In this paper we prove that if ▫$X$▫ is a complete metrizable space and ▫$\dim X \le n$▫, ▫$wX \le \tau$▫, then there is a closed embedding of ▫$X$▫ into ▫$L_n(\tau)$▫. Furthermore, any map ▫$f: X \to {\mathcal{J}}(\tau)^{n+1}$▫ can be approximated arbitrarily close by a closed embedding ▫$\psi: X \to L_n(\tau)$▫. Also, relative and pointed versions are obtained. In the separable case an analogous result is obtained, in which the classic triangular Sierpiński curve (homeomorphic to ▫${\mathcal{J}}(3)$▫) is used instead of ▫${\mathcal{J}(\aleph_0)}$▫.
Secondary keywords:	matematika;topologija;dimenzija pokrivanja;posplošena krivulja Sierpińskega;univerzalni prostor;Lipscombov univerzalni prostor;vložitev;razširitev;poln metrični prostor;zaprta vložitev;
URN:	URN:SI:UM:
Type (COBISS):	Not categorized
Pages:	str. 1-14
Volume:	ǂVol. ǂ44
Issue:	ǂšt. ǂ1009
Chronology:	2006
ID:	66868