Z = cly ( D ' ) C • f - 1 ( A )

We give an example of a nonsupercompaet continuous image of a supercompact space. 0. Introduction. This paper deals with supercompact spaces. A space is called supercompact (cf. de Groot [7] ) provided it has a closed subbase such that any of its linked subsystems (a system of sets is called linked if any two of its members meet) has nonempty intersection. Much work has been done to show that certain spaces are supercompact and that certain spaces are not supercompact. We want to mention Strok & Szyma'nski [11], who showed that all compact metric spaces are supercompact (easier proofs are available now, see van Douwen [4] and Mills [9] ) and Bell [2] who gave the first examples of compact (Hausdorff) spaces which are not supercompact. Another big class of supercompact spaces has recently been discovered. Mills [ 10] has shown that every compact topological group is supercompact. It has been open for some time whether every dyadic space (i.e. a space which is a continuous image of a family of two point discrete spaces) is supercompact. Notice that every compact topological group is dyadic (cf. Ku,z'mi•ov [8]) so that Mills' result stated above gives a partial answer to this question. Since every dyadic space is the continuous image of a supercompact space, the question arises whether the continuous image of a supercompact space is supercompact (cf. van Douwen & van Mill [6]). We will give an example of a nonsupercompact space which is a continuous image of a supercompact space (all our spaces are Hausdorff; a T 1 example was earlier given by Verbeek [12] .). In addition we will prove a theorem which as an application allows us to give a surprisingly simple proof that a space like •co is not