Algebraic and Geometrical Methods in Topology by L.F. McAuley

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By L.F. McAuley

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Very intersting applications are given in [M49] and [M52], where the Baire Category Theorem is used in the hyperspace of the unit interval to investigate some special Cantor sets playing a vital role in the theory of Fourier Series. In the second paper, Mazurkiewicz proves the existence of a Cantor set of logarithmic capacity null, which is not a countable union of compact H-sets, introduced by Raichman. e. U-set, is a countable union of compact H-sets. Recently, Debs and Saint-Raymond proved that for any Borel collection B of compact U-sets, closed with respect to compact subsets, there is a compact U-set which is not a countable union of elements of B and, as Mazurkiewicz already noticed in his first paper, Hsets form a Gb -set in the hyperspace (see [9]).

M59] FM 31, pp. 247-258. [M60] FM 33, pp. 177-228. , Theorie des Operations Lineaires, Warszawa 1932. , Topologie generale, Chap. 9, Paris 1958. , Fixed Point Theory, Warszawa 1982. , Theory of Dimensions Finite anf Infinite, Heldermann Verlag 1995. , Theorie der reellen Funktionen, Berlin 1921. , Grundriss der Mengerschen Dimensionstheorie, Math. Ann. 98 (1928), 64-88. [9] Kechris, AS. , Descriptive Set Theory and the Structure of Sets of Uniqueness, Cambridge 1987. , A half century of Polish Mathematics, Remembrances and Reflections, Warszawa 1980.

Let us illustrate the approach by examining the main idea of the first paper. Mazurkiewicz proves there that, in the hyperspace of the cube, locally connected continua form a set of exactly class Fj. To this end he associates to an arbitrary F,5-set A in the interval I a compactum K(A) in I3 and an open map f : K(A) -+ I such that f -1(t) is an arc if t c A, and a non-locally connected continuum if t E I \ A. It follows that the descriptive complexity of the set of locally connected continua in the hyperspace of K(A) is the same as the complexity of A.

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