NONCOMPACT ABSOLUTE EXTENSORS IN DIMENSIONn,n-SOFT MAPPINGS, AND THEIR APPLICATIONS

Locally convex linear topological spaces that are homeomorphic to the powers of the real line

“…It follows from [3] that in the class of compact spaces as well as in the class of Polish spaces the above definitions agree with the common definitions of the corresponding concepts. ^£(«)-spaces as well as «-soft mappings between them can be characterized in various ways.…”

“…By [10] we can conclude that the spectrum S x S is also factorizing. Let us additionally note that each 0-soft mapping is surjective [3]. Denote by p: E x E -► E a continuous law of addition that is given in E as in the linear topological space.…”

“…Good examples of results of this theory are Shchepin's theorems concerning topological characterizations of uncountable powers of two-point discrete space and of the closed unit interval [10]. At the same time it should be observed that the standard definition of the concept of an AE(0)-space is satisfactory only for various subclasses of the class of normal spaces (see [3] for the complete discussion of corresponding reasons). This made it impossible to use this notion in studying nonmetrizable linear topological spaces and the associated questions of infinite-dimensional topology and functional analysis.…”

“…Moreover, in [3] topological characterizations of uncountable powers of the real line and of the space of natural numbers were also obtained (let us recall that countable infinite powers of the above spaces were characterized correspondingly by Toruñczyk [11] and by Alexandrov and Urysohn [1]). Let us mention only one proposition [3] concerning arbitrary >4£'(0)-spaces (all exact definitions will be given below) that, together with Theorem A, can be considered as a key to a solution of our problem: the class of Polish spaces coincides with the class of all ^£'(0)-spaces with countable weight. Now we can formulate the main result of this paper which, together with Theorem A, gives a complete topological characterization of those locally convex linear spaces which are homeomorphic to powers of the line.…”

“…In the class of compact spaces the following notions were defined by Shchepin [10]. In the present form they were given in [3]. Defintion 1.…”