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Mal’tsev and retral spaces. (English) Zbl 0888.54037
In 1954, A. I. Mal’tsev [Transl., II. Ser., Am. Math. Soc. 27, 125-142 (1963); translation from Mat. Sb., n. Ser. 35(77), 3-20 (1954; Zbl 0057.02403)] introduced and studied the spaces \(X\) for which there exists a continuous mapping \(M:X^3\to X\) such that \(M(x,y,y)= x= M(y,y,x)\) for all \(x,y\in X\). Such spaces are called Mal’tsev spaces. It is known that every retract of a topological group (\(\equiv\) a retral space) is Mal’tsev. The authors answer the old question whether every retral space is Mal’tsev in the negative by presenting a counterexample making use of free topological groups. It is also shown in the article that there exists a Lindelöf topological group with cellularity equalling the continuum, thus answering (in the negative) a question posed by the reviewer [Czech. Math. J. 34(109), 541-551 (1984; Zbl 0584.22001)].

MSC:
54H11 Topological groups (topological aspects)
22A05 Structure of general topological groups
54D30 Compactness
54D70 Base properties of topological spaces
54D65 Separability of topological spaces
54E35 Metric spaces, metrizability
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