Product Embedding Methods for Free (Para)topological groups
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Abstract
Product embedding methods constitute one of the fundamental tech- niques in the construction of free topological groups and in the study of their universal properties. In this paper, we investigate the role of these methods in the construction of free topological and free paratopo- logical groups. Beginning with the classical constructions of Kakutani and Hewitt–Ross, we analyze the use of separating families of continuous mappings, evaluation maps, diagonal embeddings, and product represen- tations as the main ingredients in the existence and structural character- ization of free objects. We then examine Elfard’s adaptation of this ap- proach to free paratopological groups, where the absence of continuous in- version requires substantial modifications of the classical arguments while preserving the universal mapping property. A comparison between the topological and paratopological constructions is given, highlighting both the common embedding framework and the essential differences arising from the weaker algebraic-topological assumptions. The analysis demon- strates that product embeddings provide a unifying mechanism for free (para)topological group constructions and suggests possible extensions of these methods to categories with weaker continuity conditions, such as quasitopological and semitopological groups.
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