Communications of the
Korean Mathematical Society CKMS

Using the countably way below relation, we show that the category {\bf $\sigma$CFrm} of $\sigma$-coherent frames and $\sigma$-coherent homomorphisms is coreflective in the category {\bf Frm} of frames and frame homomorphisms. Introducing the concept of stably countably approximating frames which are exactly retracts of $\sigma$-coherent frames, it is shown that the category {\bf SCAFrm} of stably countably approximating frames and $\sigma$-proper frame homomorphisms is coreflective in {\bf Frm}. Finally we introduce strongly Lindel\"of frames and show that they are precisely lax retracts of $\sigma$-coherent frames.

Keywords: frames, countably approximating frames, $\sigma$-frames, $\sigma$-coherent frames, stably countably approximating frames

MSC numbers: 06A99, 54A99, 54D99