Functors and uncountable powers of compacta

Diffeomorphism groups of non-compact manifolds endowed with the WhitneyC∞-topology

“…The converse is not true as shown by the example of the hyperspace exp 2 ({0, 1} ℵ2 ) which is Milutin but not Dugundji, see [9, 6.7]. Theorem 1.1 shows that Dugundji compact spaces are tightly connected with the functor of probability measures P (this was observed and widely exploited byŠčepin in [22]). The relations of the class of Dugundji spaces to some other functors was studied by Alkinson and Valov [1], [25].…”

“…This fact was crucial in the proof of the coincidence of the class OG of openly generated compacta with the class κM of κ-metrizable compacta, see [22]. According to [21] and [22], κ-metrizable compacta can be characterized as follows:…”

“…We shall say that a functor F : Comp → Comp preserves preimages (over points) if for any surjective map f : X → Y between compact spaces and any closed (one-point This problem is not new and has been considered in [22], [4], [2], [25]. An information about the classes AR [F ] can be helpful because of the following simple fact.…”

On paracompact spaces and projectively inductively closed functors

“…All spaces are assumed to be Tychonoff, and all mappings, continuous. Any additional information on general topology and covariant functors one can find, for example, in ([8], [9], [17]). …”

“…Recall that a covariant functor F : Comp → Comp is said to be normal [17] if it satisfies the following properties:…”