The existence of free ultrafilters on ω does not imply the extension of filters on ω to ultrafilters. Issue 4 (5th August 2013) - Hall, Eric J.; Keremedis, Kyriakos; Tachtsis, Eleftherios
<x xml:space="preserve">Abstract</x> Let X be an infinite set and let <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z3d2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />BPI(X)</alternatives> and <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2zc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />UF(X)</alternatives> denote the propositions "every filter onXcan be extended to an ultrafilter" and "Xhas a free ultrafilter", respectively. We denote by <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2w8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />S(X)</alternatives> the Stone space of the Boolean algebra of all subsets of X. We show: For every well‐ordered cardinal number ℵ, <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2vq" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />UF</alternatives>(ℵ) iff <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2sm" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />UF</alternatives>(2ℵ). <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z379" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />UF(ω)</alternatives> iff "<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z356" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />2ω</alternatives>is a continuous image of<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z34n" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />S(ω)</alternatives>" iff "<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z310" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />S(ω)</alternatives>has a free open ultrafilter " iff "every countably infinite subset of<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2bf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />S(ω)</alternatives>has a limit point". <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2dj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />BPI(ω)</alternatives> implies "every open filter on<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z2c0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />2ω</alternatives>extends to an open ultrafilter" implies "<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z1x9" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />2ω</alternatives>has an open ultrafilter" implies <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z1tn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />UF(ω).</alternatives> It is relatively consistent with <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z221" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />ZF</alternatives> that <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z20x" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />UF</alternatives>(ω) holds, whereas <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z267" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />BPI</alternatives>(ω) fails. In particular, none of the statements given in (2) implies <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg2pr0z244" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" />BPI</alternatives>(ω).