On c-completely regular frames

Motivated by definitions of countable completely regular spaces and completely below relations of frames, we define what we call a c-completely below relation, denoted by \prec\!\!\prec_c, in between two elements of a frame. We show that a\prec\!\!\prec_c b for two elements a, b of a frame L if and only if there is \alpha\in\mathcal{R}L such that \coz\alpha\wedge a=۰ and \coz(\alpha-{\bf۱})\leq b where the set \{r\in\mathbb{R} : \coz(\alpha-{\bf r})\ne ۱\} is countable. We say a frame L is a c-completely regular frame if a=\bigvee \limits_{x\prec\!\!\prec_ca}x for any a\in L. It is shown that a frame L is a c-completely regular frame if and only if it is a zero-dimensional frame. An ideal I of a frame L is said to be c-completely regular if a\in I implies a\prec\!\!\prec_c b for some b\in I. The set of all c-completely regular ideals of a frame L, denoted by {\mathrm{c-CRegId}}(L), is a compact regular frame and it is a compactification for L whenever it is a c-completely regular frame. We denote this compactification by \beta_cL and it is isomorphic to the frame \beta_۰L, that is, Stone-Banaschewski compactification of L. Finally, we show that open and closed quotients of a c-completely regular frame are c-completely regular.

کلیدواژه ها:

frame ، c-completely regular frame and space ، c-completely below relation ، c-completely regular ideals ، zero-dimensional frame ، compactification of frame