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Topological aspects of order in C(X)
Kandić, Marko (Author), Vavpetič, Aleš (Author)

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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 ▫$C_b(X)$▫.

Language:English
Keywords:vector lattices, continuous functions, separation axioms, bands and projection bands, order continuity, un-convergence
Tipology:1.01 - Original Scientific Article
Organization:FMF - Faculty of Mathematics and Physics
Year:2019
Number of pages:str. 617-635
Numbering:Vol. 23, iss. 3
UDC:517.982.22:515.122
ISSN on article:1385-1292
DOI:10.1007/s11117-018-0628-8 Link is opened in a new window
COBISS.SI-ID:18551897 Link is opened in a new window
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Downloads:118
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Record is a part of a journal

Title:Positivity
Shortened title:Positivity
Publisher:Kluwer Academic Publishers
ISSN:1385-1292
COBISS.SI-ID:512122649 This link opens in a new window

Secondary language

Language:Slovenian
Title:Topološki vidiki urejenosti v C(X)
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

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