Estimations of small transfinite dimension in separable metrizable spaces
by
V. A. Chatyrko
Linköping
Coauthors: Y. Hattori (Matsue)

We improve some known inequalities (cf. [1]) describing mutual relation between small transfinite dimension and transfinite dimension D in separable metrizable spaces. For example,

Let X be a space and D(X) = \alpha >= \omega₀. Then trind X <= \lambda(\alpha) + m +1, where m is an integer such that 0 <= n(\alpha) +1 <= 2^m -1.

We also estimate small transfinite dimension of a product with a finite-dimensional factor, generalizing the results due to Luxemburg [2]. For example,

Let X be an infinite-dimensional compact space with trind X = \alpha. Let also the subspace F = X \{ x in X : there exists an open neighborhood Ox of x with trind Ox < \lambda(\alpha) } of X be finite-dimensional. Then there exists an integer k(ind F) such that trind (X x Y) < trind X + ind Y for any finite dimensional space Y with ind Y >= k(ind F).

References

[1] R. Engelking, Theory of Dimensions, Finite and Infinite, Heldermann Verlag, Berlin, 1995.

[2] L. A. Luxemburg, On compact metric spaces with noncoinciding transfinite dimensions, Pacific J. Math. 93 (1981) 339-386.

Date received: June 15, 2000