The Boolean prime ideal theorem and products of cofinite topologies. Issue 6 (November 2013)

• Record Type: Journal Article
• Authors: Keremedis, Kyriakos
• Is Part Of: Mathematical logic quarterly. Volume 59:Issue 6(2013)
• Page Start: 382
• Page End: 392
• Publication Date: 2013-11
• Subjects: Mathematics -- Periodicals; Logic, Symbolic and mathematical -- Periodicals; 511.3
• DOI: 10.1002/malq.201100077
• Languages: English
• ISSNs: 0942-5616

Abstract

<abstract abstract-type="main"> <title> <x xml:space="preserve">Abstract</x> </title> <p>We show: <list id="malq201100077-list-0001" list-type="roman-lower"><list-item><p>The Boolean Prime Ideal theorem <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq37s" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0001" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="sans-serif">BPI</mml:mi></mml:math></alternatives> is equivalent to each one of the statements:</p><p><list id="malq201100077-list-0002" list-type="alpha-lower"><list-item><p>"<italic>For every family</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq38b" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0002" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi mathvariant="bold">X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>=</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>X</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>, </mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>)</mml:mo></mml:mrow><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:msub></mml:math></alternatives><italic>of compact spaces, for every family</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq39w" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0003" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">G</mml:mi><mml:mspace width="0.33em" /><mml:mo>=</mml:mo><mml:mo>{</mml:mo><mml:mo>⋃</mml:mo><mml:mrow><mml:mo>{</mml:mo><mml:msubsup><mml:mi>π</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>}</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>j</mml:mi><mml:mo>∈</mml:mo><mml:mi>J</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives><italic>of basic closed sets of the product</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq3bf" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0004" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="bold">X</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mo>∏</mml:mo><mml:mrow><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>I</mml:mi></mml:mrow></mml:msub><mml:msub><mml:mi mathvariant="bold">X</mml:mi><mml:mi>i</mml:mi></mml:msub></mml:mrow></mml:math></alternatives><italic>with the fip there is a family of subbasic closed sets</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq3c0" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0005" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="script">H</mml:mi></mml:math></alternatives> (<alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq3dj" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0006" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="script">H</mml:mi><mml:mo>⊂</mml:mo><mml:mo>{</mml:mo><mml:msubsup><mml:mi>π</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:mi>F</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>i</mml:mi><mml:mo>∈</mml:mo><mml:mi>I</mml:mi><mml:mo>, </mml:mo><mml:msup><mml:mi>F</mml:mi><mml:mi mathvariant="normal">c</mml:mi></mml:msup><mml:mo>∈</mml:mo><mml:msub><mml:mi>T</mml:mi><mml:mi>i</mml:mi></mml:msub><mml:mo>}</mml:mo></mml:mrow></mml:math></alternatives>) <italic>with the fip such that for every</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq3f3" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0007" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>j</mml:mi><mml:mo>∈</mml:mo><mml:mi>J</mml:mi><mml:mo>, </mml:mo><mml:mi mathvariant="script">H</mml:mi><mml:mo>∩</mml:mo><mml:mo>{</mml:mo><mml:msubsup><mml:mi>π</mml:mi><mml:mrow><mml:mi>i</mml:mi></mml:mrow><mml:mrow><mml:mo>−</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msubsup><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mrow><mml:mi>i</mml:mi><mml:mi>j</mml:mi></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow><mml:mo>:</mml:mo><mml:mi>q</mml:mi><mml:mo>∈</mml:mo><mml:msub><mml:mi>Q</mml:mi><mml:mi>j</mml:mi></mml:msub><mml:mo>}</mml:mo><mml:mo>≠</mml:mo><mml:mi>⌀</mml:mi></mml:mrow></mml:math></alternatives>".</p></list-item><list-item><p>"<italic>For every compact Loeb space</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq3gn" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0008" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="bold">Y</mml:mi></mml:math></alternatives><italic>(the family of all non empty closed subsets of</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq3h6" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0009" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="bold">Y</mml:mi></mml:math></alternatives><italic>has a choice function) and for every set</italic><italic>X</italic><italic>the product</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq4d2" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0010" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi mathvariant="bold">Y</mml:mi><mml:mi>X</mml:mi></mml:msup></mml:math></alternatives><italic>is compact"</italic>.</p></list-item></list></p></list-item><list-item><p><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq49d" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0011" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="sans-serif">AC</mml:mi><mml:mi> fin </mml:mi></mml:msub></mml:math></alternatives> (: the axiom of choice restricted to families of finite sets) implies "<italic>every well ordered product of cofinite topologies is compact</italic>" and "<italic>every well ordered basic open cover of a product of cofinite topologies has a finite subcover</italic>".</p></list-item><list-item><p><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq4bz" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0012" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi mathvariant="sans-serif">CAC</mml:mi><mml:mi> fin </mml:mi></mml:msub></mml:math></alternatives> (: the axiom of choice restricted to countable families of finite sets) iff "<italic>every countable product of cofinite topologies is compact</italic>".</p></list-item><list-item><p><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq479" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0013" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi mathvariant="sans-serif">BPI</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> (: every filter of <alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq48v" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0014" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>℘</mml:mi><mml:mo>(</mml:mo><mml:mi>ω</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:math></alternatives> extends to an ultrafilter) is equivalent to the proposition "<italic>for every compact Loeb space</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq4mc" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0015" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi mathvariant="bold">Y</mml:mi></mml:math></alternatives><italic>having a base of size</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq4pg" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0016" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>≤</mml:mo><mml:mo>|</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math></alternatives><italic>and for every set</italic><italic>X</italic><italic>of size</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq4j8" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0017" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mo>≤</mml:mo><mml:mo>|</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:math></alternatives><italic>the product</italic><alternatives><inline-graphic mimetype="image" xlink:href="ark:/27927/pgg3wjhq4kt" xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" /><mml:math display="inline" altimg="urn:x-wiley:09425616:malq201100077:equation:malq201100077-math-0018" overflow="scroll" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mi mathvariant="bold">Y</mml:mi><mml:mi>X</mml:mi></mml:msup></mml:math></alternatives><italic>is compact</italic>".</p></list-item></list></p> </abstract>