The point space of a frame

sends a frame A to its point space pt(A). This is an important construction which enables quite a lot of point-sensitive topology, that is point set topology, to be done in a point-free way, that is using frames. In Section 1 we first set up the point space and the associated adjunction in what may seem to be a rather ad hoc fashion. After that, in Section 2, we show that the adjunction is schizophrenically induced (by a rather trivial object). This explains much of the behaviour of the adjunction, and brings out many of its other features. You may prefer to read that section first before reading the ad hoc version. In Section 3 we give an entirely point-sensitive account of the construction. This, in fact, was the original version of the construction, and again you might want to read it before either of the first two sections. We then look at two particular examples of the point-space construction. In Section 4 we look at the ideal completion of a poset, and in Section 5 we look at the Stone representation of distributive lattices. These two section can be read in either order. The point space construction depends on attaching a set pt(A) of points to a frame A. In can happen that pt(A) is empty even though A is quite large and complicated. Section 6 gives an example of such an exotic frame. On the other hand, many frames have enough points to be a topology in disguise. Section 7 considers this aspect. This involves the use of variants of Zorn’s Lemma. Again these last two section can be read in either order.