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Souslin's conjecture and some equivalences Daniel, Ian Alistair

Abstract

This thesis deals with some equivalences to Soulin's hypothesis. Also included is a consequence of its negation, the existence of a normal Hausdorff space which is not countably paracompact. Two equivalences to Souslin's hypothesis can be obtained by considering the countable chain condition in the product with itself of an ordered continuum and by considering Peano maps on ordered continua. An equivalence to a negative answer is the existence of a tree of cardinality, [formula omitted] ₁, in which every chain and antichain (set of pairwise incomparable objects) is countable. Another equivalence to Souslin's hypothesis is obtained by assuming that a continuum which is the continuous image of an ordered continuum is metrizable if and only if it has the countable chain condition. We get yet another by assuming that a compactum which is the continuous image of an ordered compactum is separable if and only if it has the countable chain condition. Finally, if we restrict ourselves to compacta of the type just mentioned we find that Souslin's hypothesis is equivalent to a generalization of a metrization theorem of S. Mardeśic.

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