J. Aust. Math. Soc. 83 (2007), no. 1, pp. 135–148.

Local homomorphisms of topological groups

 Yevhen Zelenyuk School of MathematicsUniversity of the WitwatersrandPrivate Bag 3Wits 2050South Africazelenyuk@maths.wits.ac.za
Received 21 November 2005; revised 31 May 2006
Communicated by G. Willis

Abstract

A mapping f:G\to S from a left topological group G into a semigroup S is a local homomorphism if for every x\in G\setminus \{e\}, there is a neighborhood U_x of e such that f(xy)=f(x)f(y) for all y\in U_x\setminus \{e\}. A local homomorphism f:G\to S is onto if for every neighborhood U of e, f(U\setminus \{e\})=S. We show that

1. every countable regular left topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto \mathbb {N};
2. it is consistent that every countable topological group containing a discrete subset with exactly one accumulation point admits a local homomorphism onto any countable semigroup;
3. it is consistent that every countable nondiscrete maximally almost periodic topological group admits a local homomorphism onto the countably infinite right zero semigroup.

 2000 Mathematics Subject Classification: primary 22A30, 54H11; secondary 54A35, 54G05 (Metadata: XML, RSS, BibTeX) MathSciNet: MR2354??? Z'blatt-MATH: pre05231337

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