Abstract. We show that there is no monad based on the normal functor H introduced earlier by Radul which is a certain functorial compactification of the HartmanMycielski construction HM . IntroductionThe general theory of functors acting on the category Comp of compact Hausdorff spaces (compacta) and continuous mappings was founded by Shchepin [Sh]. He described some elementary properties of such functors and defined the notion of the normal functor which has become very fruitful. The classes of all normal functors include many classical constructions: the hyperspace exp, the space of probability measures P , the space of idempotent measures I, and many other functors (cf.[ FZ], [TZ], [Z]).Let X be a space and d an admissible metric on X bounded by 1. By HM (X) we shall denote the space of all maps from [0, 1) to the space X such that f |[t i , t i+1 ) ≡ const, for some 0 = t 0 ≤ · · · ≤ t n = 1, with respect to the following metricThe construction of HM (X) is known as the Hartman-Mycielski construction [HM] and was introduced for purposes of topological groups theory. However it found some applications not connected with groups (see for example [Z1]).The construction HM was considered for any compactum Z in [TZ; 2.5.2]. Let U be the unique uniformity of Z. For every U ∈ U and ε > 0, letThe sets < α, U, ε > form a base of a topology in HM Z. The construction HM acts also on maps. Given a map f : X → Y in Comp, define a map HM X → HM Y by the formula HM F (α) = f • α. In general, HM X is not compact. Let us fix some n ∈ N. For every compactum Z consider HM n (Z) = f ∈ HM (Z) | there exist 0 = t 1 < · · · < t n+1 = 1 with f |[t i , t i+1 ) ≡ z i ∈ Z, i = 1, . . . , n .1991 Mathematics Subject Classification. 54B30, 57N20.
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