Collapsing algebra

<h2 id="definition">Definition</h2>
<p>There are several slightly different sorts of collapsing algebras. 
</p><p>If κ and λ are cardinals, then the Boolean algebra of <a href="/facts/Regular_open_set/AoX6h3w4">regular open sets</a> of the <a href="/facts/Product_space/0hg02IaX">product space</a> κλ is a collapsing algebra. Here κ and λ are both given the <a href="/facts/Discrete_topology/hpvXA23u">discrete topology</a>. There are several different options for the topology of 
κλ. The simplest option is to take the usual product topology. Another option is to take the topology generated by open sets consisting of functions whose value is specified on less than λ elements of λ.
</p>

<ul><li>Bell, J. L. (1985). <a href="https://archive.org/details/booleanvaluedmod0000bell"><i>Boolean-Valued Models and Independence Proofs in Set Theory</i></a>. Oxford Logic Guides. Vol. 12 (2nd ed.). Oxford: Oxford University Press (Clarendon Press). <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-19-853241-5. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0585.03021">0585.03021</a>.</li>
<li><a href="/facts/Thomas_Jech/E4JQu4Mk">Jech, Thomas</a> (2003). <i>Set theory</i> (third millennium (revised and expanded) ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 3-540-44085-2. <a href="/facts/OCLC_(identifier)/Yqad8waq">OCLC</a> <a href="https://search.worldcat.org/oclc/174929965">174929965</a>. <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:1007.03002">1007.03002</a>.</li>
<li><a href="/facts/Azriel_L%25C3%25A9vy/SCxxPAFn">Lévy, Azriel</a> (1963). "Independence results in set theory by Cohen's method. IV". <i>Notices Amer. Math. Soc</i>. 10.</li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Lévy 1963, p. 593. - Lévy, Azriel (1963). "Independence results in set theory by Cohen's method. IV". Notices Amer. Math. Soc. 10. <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
</ol>

Collapsing algebra open-in-new

Collapsing algebra