Localic Remote Points Revisited

Themba Dube, Martin M. Mugochi

We consider remote points in general extensions of frames, with an emphasis on perfect extensions. For a strict extension $\tau_\mathfrak{X}L\mapsto L$ determined by a set $\mathfrak X$ of filters in $L$, we show that if there is an ultrafilter in $\mathfrak X$ then the extension has a remote point. In particular, if a completely regular frame $L$ has a maximal completely regular filter which is an ultrafilter, then $\beta L\to L$ has a remote point, where $\beta L$ is the Stone-Čech compactification of $L$. We prove that in certain extensions associated with radical ideals and $\ell$-ideals of reduced $f$-rings, remote points induced by algebraic data are exactly non-essential prime ideals or non-essential irreducible $\ell$-ideals. Concerning coproducts, we show that if $M_1\to L_1$ and $M_2\to L_2$ are extensions of $T_1$-frames, then each of these extensions has a remote point if the extension $M_1\otimes M_2\to L_1\otimes L_2$ has a remote point.