Normality for biframes
A frame is a complete lattice \(L_ 0\) with \(a\wedge (\bigvee X)= \bigvee_{x\in X} (a\wedge x)\), \(a\in L_ 0\), \(X\subseteq L_ 0\). A frame map is a homomorphism which preserves arbitrary joins, 0, \(e\in L_ 0\) (\(e\) unit element). A biframe is a triple \(L= (L_ 0, L_ 1, L_ 2)\), \(L_ i\subseteq L_ 0\) are subframes of the frame \(L_ 0\), \(i= 1,2\), and \(L_ 0\) is generated by \(L_ 1\cup L_ 2\). A homomorphism of \(L\) is a homomorphism \(h\) of \(L_ 0\) with \(h(L_ i)\subseteq M_ i\), \(i= 1,2\) into the biframe \((M_ 0, M_ 1, M_ 2)\). \(L\) is called normal if for \(x\in L_ i\), \(y\in L_ k\), \(x\vee y= e\), there exist \(u\in L_ k\), \(v\in L_ i\) with \(u\wedge v= 0\), \(x\vee u= e= y\vee v\). A bispace is a triple \((X, P, Q)\), where \(P\), \(Q\) are topologies on the set \(X\). There exists a functor \(O\) from bispaces to biframes. The author proves that a biframe \(L\) is normal iff the following conditions hold: for \(x\in L_ 1\), \(y\in L_ 2\) with \(x\vee y= e\) there exists a biframe homomorphism \(\mu: O\mathbb{R}\to L\) with \(\mu((- \infty, 1))\leq x\), \(\mu((0, \infty))\leq y\). This is an extended version for biframes of Urysohn's lemma. Normal biframes include biframes which are compact regular, Lindelöf regular, strictly 0-dimensional, Boolean, for which some additional results are shown.
scientific article; zbMATH DE number 740361
scientific article