Vol. 20, No. 3, 1967

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Quasi dimension type. I. Types in the real line

Jack Segal

Vol. 20 (1967), No. 3, 501–534

Fréchet defined the concept of dimension type in attempting to obtain a reasonable way of comparing two abstract topological spaces. The Fréchet dimension type of a topological space X is said to be less than or equal to the Fréchet dimension type of the topological space Y if and only if there is a homeomorphism between X and a subset of Y . In this case we write dX dY . Notice that the statement, dX dY , is equivalent to the statement, X can be (topologically) embedded in Y .

Fréchet dimension type also is a more delicate way of comparing two spaces (when it applies) than covering dimension (denoted by dim). One difficulty is that many spaces are not comparable with respect to Fréchet dimension type because of the strong restriction of requiring that one be embeddable in the other. In this paper we will relax this restriction somewhat to obtain a new dimension type called quasi dimension type. Under quasi dimension type many more spaces are comparable and yet many of the properties of Fréchet dimension type are retained. Kuratowski showed that there are 2e Fréchet dimension types represented by subsets of the real line. We will show that there are only denumerably many quasi dimension types represented by subsets of the real line- Furthermore, we completely determine the partial ordering of these types and give a topological characterization of the linear sets having a given type.

Mathematical Subject Classification
Primary: 54.70
Received: 20 August 1963
Published: 1 March 1967
Jack Segal