Diagonal-flexible spaces and rotoids

This article deals with several non-standard generalizations of the classical concept of a topological group. An important common feature of these generalizations is the fact that all of them are given in geometric terms. They are based on the concept of a diagonal-flexible space. These spaces were introduced and studied in [6] under the slightly different name of a diagonal resolvable space.We provide a brief survey of results obtained so far in this direction, and also obtain some new results on the structure of rotoids which are very close to diagonal-flexible spaces. One of the main new results below is Theorem 3.1: Every compact rotoid of countable tightness is metrizable. Using this fact, we establish that if X is a compact hereditarily normal strong rotoid, then X is metrizable (Theorem 3.5). The two theorems just mentioned suggest that compact rotoids strongly resemble, by their topological properties, compact topological groups and, more generally, dyadic compacta. Several open problems on rotoids are mentioned, in particular, the next one: is every compact rotoid a dyadic compactum?

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