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In [3] we constructed a contravariant functor Top

By Harold Simmons


from spaces to frames. The object assignment sends a space S to its topology OS, and the arrow assignment sends a map φ to the inverse image function φ ← (restricted to the topologies). In this document we show that this functor is one half of a contravariant adjunction between the two categories. The object assignment in the other direction A ✲ pt(A) 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 versio

Year: 2006
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