The Aleksandrov-Urysohn conjecture about the cardinality of a first countable compact spare X is here given an equivalent formulation in terms of a generalized Lindelöf condition in the weak topology on X generated by the Baire functions. A number of related conditions are shown to be equivalent (without assuming that X is first countable); these include two sequential properties of the pointwise topology on various spaces of real-valued functions on X, and the condition that X is dispersed (i.e., has no non-void perfect subsets).
Added in prof: The Alexandrov-Urysohn conjecture has been generalized and proved by A. V. Archangel' skii (On the cardinality of bicompacta satisfying the first axiom of countability, Dokl. Akad. Nauk SSSR 187 (1969) (Russian); translated as Soviet Math. Dokl. 10 (1969) pp. 951–955). In his more general theorem the compactness hypothesis is weakened to Lindelöf.
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Dedicated to the sixtieth birthday of Prof. Edgar R. Lorch
This research was partially supported by the National Science Foundation.
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Meyer, P.R. Function spaces and the aleksandrov-urysohn conjecture. Annali di Matematica 86, 25–29 (1970) doi:10.1007/BF02415705
- Nauk SSSR
- Function Space
- Related Condition
- Equivalent Formulation
- Weak Topology