Topological aspects of order in C(X)

Marko Kandić (Author), Aleš Vavpetič (Author)

Abstract

Obravnavamo povezavo med urejenostjo in topologijo v mreži ▫$C(X)$▫ zveznih funkcij na Hausdorffovem prostoru ▫$X$▫. Dokažemo, da zožitev funkcije ▫$f\in C(X)$▫ na zaprto podmnožico ▫$A$▫ v primeru, ko je ▫$X\in T_{3\frac{1}{2}}$▫, inducira urejenostno zvezen operator natanko takrat, ko je ▫$A=\overline{\mathrm{Int\,}A}$▫. S tem rezultatom na enostaven način karakteriziramo pasove in projekcijske pasove v ▫$C_0(X)$▫, ▫$C_b(X)$▫ in ▫$C(X)$▫. Z dobljenimi rezultati podamo pozitiven odgovor na vprašanje o dvigu un-konvergence z zaprtih idealov mrež ▫$C_0(X)$▫ in ▫$C_b(X)$▫.

Keywords

vektorske mreže;zvezne funkcije;separacijski aksiomi;pasovi in projekcijski pasovi;urejenostna zveznost;un-konvergenca;vector lattices;continuous functions;separation axioms;bands and projection bands;order continuity;un-convergence;

Data

Language:	English
Year of publishing:	2019
Typology:	1.01 - Original Scientific Article
Organization:	UL FMF - Faculty of Mathematics and Physics
UDC:	517.982.22:515.122
COBISS:	18551897
ISSN:	1385-1292
Views:	441
Downloads:	240
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	Slovenian
Secondary title:	Topološki vidiki urejenosti v C(X)
Secondary abstract:	In this paper we consider the relationship between order and topology in the vector lattice ▫$C(X)$▫ of all continuous functions on a Hausdorff space ▫$X$▫. We prove that the restriction of ▫$f\in C(X)$▫ to a closed set ▫$A$▫ in the case when ▫$X\in T_{3 \frac 12}$▫ induces an order continuous operator iff ▫$A= \overline{\mathrm{Int\,}A}.$▫ This result enables us to easily characterize bands and projection bands in ▫$C_0(X)$▫, ▫$C_b(X)$▫ and ▫$C(X)$▫. Our results serve us to provide a positive answer to the question on lifting un-convergence from closed ideals of ▫$C_0(X)$▫ and ▫$C_b(X)$▫.
Secondary keywords:	vektorske mreže;zvezne funkcije;separacijski aksiomi;pasovi in projekcijski pasovi;urejenostna zveznost;un-konvergenca;
Pages:	str. 617-635
Volume:	ǂVol. ǂ23
Issue:	ǂiss. ǂ3
Chronology:	July 2019
DOI:	10.1007/s11117-018-0628-8
ID:	11807749