<p>We prove some facts about locales <i>L</i> equipped with the Scott topology <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\({\Omega }(L)\)</EquationSource> </InlineEquation>, in particular studying a canonical frame homomorphism <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\phi :{\Omega }(L)\rightarrow L\)</EquationSource> </InlineEquation> which is motivated by an application to cognitive science. Such a topological locale <i>L</i> is called a <i>Scott locale</i> if the inclusion of primes <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(p:{\Sigma }(L)\rightarrow L\)</EquationSource> </InlineEquation> is continuous. We prove that the spectrum <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\({\Sigma }(L)\)</EquationSource> </InlineEquation> of a Scott locale <i>L</i> is necessarily <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(T_1\)</EquationSource> </InlineEquation>, and that preregular locales (a generalization of regular locales) are Scott locales. If <i>L</i> is the topology of a topological space <i>X</i> we find a (necessarily unique) continuous map <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(f:X\rightarrow L\)</EquationSource> </InlineEquation> such that <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(f^{-1}=\phi \)</EquationSource> </InlineEquation> and compare it with the points-to-primes map <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(p:X\rightarrow L\)</EquationSource> </InlineEquation>, showing that <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(f=p\)</EquationSource> </InlineEquation> if and only if <i>X</i> is preregular, and that a sober space <i>X</i> is Hausdorff if and only if <i>X</i> is <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(T_1\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(f(X)\subseteq {\Sigma }(L).\)</EquationSource> </InlineEquation></p>

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Scott Locales

  • Pedro Resende,
  • João Paulo Santos

摘要

We prove some facts about locales L equipped with the Scott topology \({\Omega }(L)\) , in particular studying a canonical frame homomorphism \(\phi :{\Omega }(L)\rightarrow L\) which is motivated by an application to cognitive science. Such a topological locale L is called a Scott locale if the inclusion of primes \(p:{\Sigma }(L)\rightarrow L\) is continuous. We prove that the spectrum \({\Sigma }(L)\) of a Scott locale L is necessarily \(T_1\) , and that preregular locales (a generalization of regular locales) are Scott locales. If L is the topology of a topological space X we find a (necessarily unique) continuous map \(f:X\rightarrow L\) such that \(f^{-1}=\phi \) and compare it with the points-to-primes map \(p:X\rightarrow L\) , showing that \(f=p\) if and only if X is preregular, and that a sober space X is Hausdorff if and only if X is \(T_1\) and \(f(X)\subseteq {\Sigma }(L).\)