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Weak completions of paratopological groups. (English) Zbl 1497.22005

Summary: Given a \(T_0\) paratopological group \(G\) and a class \(\mathcal{C}\) of continuous homomorphisms of paratopological groups, we define the \(\mathcal{C}\)-semicompletion \(\mathcal{C} [G)\) and \(\mathcal{C}\)-completion \(\mathcal{C}[G]\) of the group \(G\) that contain \(G\) as a dense subgroup, satisfy the \(T_0\)-separation axiom and have certain universality properties. For special classes \(\mathcal{C} \), we present some necessary and sufficient conditions on \(G\) in order that the (semi)completions \(\mathcal{C} [G)\) and \(\mathcal{C} [G]\) be Hausdorff. Also, we give an example of a Hausdorff paratopological abelian group \(G\) whose \(\mathcal{C} \)-semicompletion \(\mathcal{C} [G)\) fails to be a \(T_1\)-space, where \(\mathcal{C}\) is the class of continuous homomorphisms of sequentially compact topological groups to paratopological groups. In particular, the group \(G\) contains an \(\omega \)-bounded sequentially compact subgroup \(H\) such that \(H\) is a topological group but its closure in \(G\) fails to be a subgroup.

MSC:

22A15 Structure of topological semigroups
54D35 Extensions of spaces (compactifications, supercompactifications, completions, etc.)
54H11 Topological groups (topological aspects)
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References:

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