Remote points and the like in pointfree topology
A point \(p\) of \(\beta X\), where \(X\) is a Tychonoff space, is called a remote point if for any nowhere dense \(D\subseteq X,\;p\notin cl_{\beta X}D\). A subset \(S\) of \(X\) is called round if whenever the closure in \(\beta X\) of a zero-set of \(X\) contains \(S\), then it is a neighborhood of \(S\). The purpose of this paper is to study these notions in the pointfree context. In the process, the author introduces N-homomorphisms and shows how the Stone extension of an N-homomorphism transfers remote points back and forth. The pointfree topology is in terms of frames. A frame is a complete lattice \(L\) satisfying \(a\wedge\bigvee S=\bigvee\{a\wedge x\mid x\in S\}\). For \(a,b\in L,\;a\prec\prec b\) means the existence of a sequence \((x_r)\) indexed by \(r\in{\mathbb Q}\cap[0,1]\) such that \(x_0=a,x_1=b\) and for every \(r,s\) with \(r<s\) there is an element \(T_{rs}\) such that \(a\wedge t_{rs}=0\) and \(b\vee t_{rs}=1\). A frame \(L\) is called completely regular if \(a=\bigvee\{x\in L\mid x\prec\prec a\}\) for all \(a\in L\). An ideal \(J\) of \(L\) is said to be completely regular if \(\forall\,x\in J\,\exists\,y\in J\,x\prec\prec y\). The frame of all completely regular ideals of a completely regular frame \(L\) is denoted by \(\beta L\). A quotient \(h:L\longrightarrow M\) is called nowhere dense if \(\forall\,x\neq0\,\exists\,y\neq0\,(y\leq x\, \&\, h(y)=0)\). A point \(I\in\beta L\) is called remote if for every nowhere dense quotient \(h\), \[ I\vee\{x\in L\mid x\prec\prec\bigvee h^{-1}(0)\}=\top. \] An N-homomorphism is a frame homomorphism \(h:L\longrightarrow M\) such that \(y\prec\prec h(x)\) implies \(y\prec\prec h(s)\) for some \(s\prec\prec x\). The notation \(\beta X\) is not explained.
scientific article; zbMATH DE number 5652014
scientific article