uu.seUppsala University Publications

CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_upper_j_idt160",{id:"formSmash:upper:j_idt160",widgetVar:"widget_formSmash_upper_j_idt160",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:upper:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_upper_j_idt163_j_idt166",{id:"formSmash:upper:j_idt163:j_idt166",widgetVar:"widget_formSmash_upper_j_idt163_j_idt166",target:"formSmash:upper:j_idt163:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});

The Vitali covering theorem in constructive mathematicsPrimeFaces.cw("AccordionPanel","widget_formSmash_some",{id:"formSmash:some",widgetVar:"widget_formSmash_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_all",{id:"formSmash:all",widgetVar:"widget_formSmash_all",multiple:true});
function selectAll()
{
var panelSome = $(PrimeFaces.escapeClientId("formSmash:some"));
var panelAll = $(PrimeFaces.escapeClientId("formSmash:all"));
panelAll.toggle();
toggleList(panelSome.get(0).childNodes, panelAll);
toggleList(panelAll.get(0).childNodes, panelAll);
}
/*Toggling the list of authorPanel nodes according to the toggling of the closeable second panel */
function toggleList(childList, panel)
{
var panelWasOpen = (panel.get(0).style.display == 'none');
// console.log('panel was open ' + panelWasOpen);
for (var c = 0; c < childList.length; c++) {
if (childList[c].classList.contains('authorPanel')) {
clickNode(panelWasOpen, childList[c]);
}
}
}
/*nodes have styleClass ui-corner-top if they are expanded and ui-corner-all if they are collapsed */
function clickNode(collapse, child)
{
if (collapse && child.classList.contains('ui-corner-top')) {
// console.log('collapse');
child.click();
}
if (!collapse && child.classList.contains('ui-corner-all')) {
// console.log('expand');
child.click();
}
}
(English)Article in journal (Refereed) Submitted
##### Abstract [en]

##### Keywords [en]

Constructive mathematics, Reverse mathematics, Measure theory, Vitali's covering theorem, Formal topology
##### Research subject

Mathematical Logic
##### Identifiers

URN: urn:nbn:se:uu:diva-152066OAI: oai:DiVA.org:uu-152066DiVA, id: diva2:412413
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt600",{id:"formSmash:j_idt600",widgetVar:"widget_formSmash_j_idt600",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt606",{id:"formSmash:j_idt606",widgetVar:"widget_formSmash_j_idt606",multiple:true});
#####

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt612",{id:"formSmash:j_idt612",widgetVar:"widget_formSmash_j_idt612",multiple:true}); Available from: 2011-04-22 Created: 2011-04-22 Last updated: 2011-06-14Bibliographically approved
##### In thesis

This paper investigates the Vitali Covering Theorem from various constructive angles. A Vitali Cover of a metric space is a cover such that for every point there exists an arbitrarily small set of the cover containing this point. The VCT now states, that for any Vitali Cover one can find a finite, disjoint family of sets in the Vitali Cover that cover the entire space up to a set of a given non-zero measure. We will show, by means of a recursive counterexample, that there cannot be a fully constructive proof, but that adding a very weak semi-constructive principle suffices to give such a proof. Lastly, we will show that with an appropriate formalization in formal topology the non-constructive problems can be avoided completely.

1. Contributions to Pointfree Topology and Apartness Spaces$(function(){PrimeFaces.cw("OverlayPanel","overlay412415",{id:"formSmash:j_idt999:0:j_idt1007",widgetVar:"overlay412415",target:"formSmash:j_idt999:0:parentLink",showEvent:"mousedown",hideEvent:"mousedown",showEffect:"blind",hideEffect:"fade",appendToBody:true});});

urn-nbn$(function(){PrimeFaces.cw("Tooltip","widget_formSmash_j_idt1827",{id:"formSmash:j_idt1827",widgetVar:"widget_formSmash_j_idt1827",showEffect:"fade",hideEffect:"fade",showDelay:500,hideDelay:300,target:"formSmash:altmetricDiv"});});

CiteExport$(function(){PrimeFaces.cw("TieredMenu","widget_formSmash_lower_j_idt1880",{id:"formSmash:lower:j_idt1880",widgetVar:"widget_formSmash_lower_j_idt1880",autoDisplay:true,overlay:true,my:"left top",at:"left bottom",trigger:"formSmash:lower:exportLink",triggerEvent:"click"});}); $(function(){PrimeFaces.cw("OverlayPanel","widget_formSmash_lower_j_idt1881_j_idt1883",{id:"formSmash:lower:j_idt1881:j_idt1883",widgetVar:"widget_formSmash_lower_j_idt1881_j_idt1883",target:"formSmash:lower:j_idt1881:permLink",showEffect:"blind",hideEffect:"fade",my:"right top",at:"right bottom",showCloseIcon:true});});