Stable Compactifications of Frames

In a classic paper, Smirnov [14] characterized the poset of compactifications of a completely regular space in terms of the proximities on the space. Later, Smyth [15] introduced the notion of a stable compactification of a T0-space and described them in terms of quasi-proximities on the space. Banaschewski [1] formulated Smirnov’s results in the pointfree setting, defining a compactification of a completely regular frame, and characterizing these in terms of the strong inclusions on the frame. We provide an alternate description of stable compactifications of T0-spaces as embeddings into stably compact spaces that are dense with respect to the patch topology, and relate such stable compactifications to ordered spaces. Each stable compactification of a T0-space induces a companion topology on the space, and we show the companion topology induced by the largest stable compactification is the topology τ studied by Salbani [11, 12]. In the pointfree setting, we introduce a notion of a stable compactification of a frame that extends Smyth’s stable compactification of a T0-space, and Banaschewski’s compactification of a frame. We characterize the poset of stable compactifications of a frame in terms of proximities on the frame, and in terms of stably compact subframes of its ideal frame. These results are then specialized to coherent compactifications of frames, and related to Smyth’s spectral compactifications of a T0-space.