Algebraic Topology, Poznan 1989 by Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski

By Stefan Jackowski, Bob Oliver, Krzysztof Pawalowski

As a part of the medical job in reference to the seventieth birthday of the Adam Mickiewicz collage in Poznan, a global convention on algebraic topology used to be held. within the ensuing complaints quantity, the emphasis is on vast survey papers, a few awarded on the convention, a few written for that reason.

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Then: l) For each a in G, the mapping )~a restricted to Gq is a homeomorphism of Gq onto itself" 2) For each a in G, flq)~aIG -- )~aflqiG; 3) )~(pq(b)) = pq(ab) = abq, for each b E G; 4) The space Gq is homogeneous. It is natural to ask if flq restricted to G is actually a homomorphism of the group G into the semigroup/3G. However, pq(a)pq(b) = aqbq and pq(ab) = abq. Since there is no reason to believe that aqbq = abq, we should not also expect that pq IG is a homomorphism. The reasoning above also shows that the subspace Gq -- {bq : b E G} is not, in general, a subgroup of the semigroup/3G.

Is every extremally disconnected (regular) paratopological group a topological group? 40. PROBLEM. Is there an example in ZFC of a nondiscrete extremally disconnected regular paratopological group? 6. Topological groups and completions A space X is called Moscow, ARHANGEL'SKII [1983], if for each open subset U of X, the closure of U in X is the union of a family of G~-subsets of X, that is, for each z E there exists a Gr-subset P of X such that z C P C U. 22 Arhangel'skii / Topological invariants in algebraic environment [Ch.

16. THEOREM (ARHANGEL' SKII [2000c1). Let G = II{Ga : a E A} be the product of topological groups Ga such that the space G is Moscow and IG] is Ulam non-measurable. Then vII{Ga : a E A} = H{vGa : a E A}. In particular, under the assumptions and in the notation of the above theorem, the formula vII{G,~ : a E A} = II{vG,~ : a E A}. T has a countable network. Indeed, if the Souslin number of the product group G is countable, then G is Moscow. This takes care of cases 1)-5). Similarly, in the cases 6), 7), and 8) the group G is also Moscow, since the 9-tightness of it is countable.

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