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.
