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Inductive Dimension of Completely Normal Spaces

In: The Mathematical Legacy of Eduard Čech

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  • C. H. Dowker

Abstract

USING the dimension defined inductively in terms of closed sets, I show that in a completely normal space the dimension of the union of two disjoint sets, one of which is closed, is at most equal to the greatest of their dimensions. A corresponding theorem is proved for a countable union of disjoint sets. It follows that in completely normal spaces the subset theorem implies the sum theorem. The subset theorem and the open subset theorem are shown to be equivalent. Therefore (Theorem 1) the sum and subset theorems hold for any completely normal space in which the dimension of a set A is never less than the dimension of a relatively open subset of A.

Suggested Citation

  • C. H. Dowker, 1993. "Inductive Dimension of Completely Normal Spaces," Springer Books, in: Miroslav Katětov & Petr Simon (ed.), The Mathematical Legacy of Eduard Čech, pages 165-177, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7524-0_14
    DOI: 10.1007/978-3-0348-7524-0_14
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