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A.V. Arhangel’skii 2AHBA?J =FFECI E JFCE?= CHKFI ?HII?FAAJ=HO IK>IAJI =@ GKJEAJI

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A.V. Arhangel’skii

Perfect mappings in topological groups, cross-complementary subsets and quotients

Comment.Math.Univ.Carolinae 44,4 (2003) 701-709.

Abstract: The following general question is considered. Suppose thatGis a topo- logical group, and F, M are subspaces of G such that G = M F. Under these general assumptions, how are the properties ofF andM related to the properties ofG? For example, it is observed that ifM is closed metrizable andF is compact, thenGis a paracompactp-space. Furthermore, ifM is closed and first countable, F is a first countable compactum, andF M =G, thenGis also metrizable. Several other results of this kind are obtained. An extensive use is made of the following old theorem of N. Bourbaki [5]: ifF is a compact subset of a topological groupG, then the natural mapping of the product spaceG×F ontoG, given by the product operation inG, is perfect (that is, closed continuous and the fibers are compact).

This fact provides a basis for applications of the theory of perfect mappings to topological groups. Bourbaki’s result is also generalized to the case of Lindel¨of sub- spaces of topological groups; with this purpose the notion of aGδ-closed mapping is introduced. This leads to new results on topological groups which areP-spaces.

Keywords: topological group, quotient group, locally compact subgroup, quo- tient mapping, perfect mapping, paracompactp-space, metrizable group, countable tightness

AMS Subject Classification: 22A05, 54H11, 54D35, 54D60

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