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Spaces of diversity one. (English) Zbl 0724.54020
A space in which every nonempty open subset is homeomorphic to the whole space is called a divine space. As basic examples serve the space of rational numbers and the space of irrational numbers with the usual topology. Several properties of divine spaces are examined and many open problems raised. Examples of results: there exist divine spaces of all infinite cardinalities. The product of two divine spaces need not be divine. If both spaces are moreover zero-dimensional and their product is hereditarily Lindelöf, then it is also divine. Every divine space is countably divisible - i.e. it can be expressed as the countably infinite union of clopen homeomorphic copies of itself.
Reviewer: J.Chvalina (Brno)

54C99 Maps and general types of topological spaces defined by maps
54A25 Cardinality properties (cardinal functions and inequalities, discrete subsets)
54D99 Fairly general properties of topological spaces
54F99 Special properties of topological spaces