Some questions of the theory of dimension of topological spaces
by
Lazar Zambakhidze
I.Javakhishvili Tbilisi State University, Tbilisi
Coauthors: Ivane Tsereteli

It is well known that the three classical dimension functions ind, Ind and dim coincide in the class T_SM of all separable metrizable spaces and their common value d=dim=Ind=ind has several nice properties among which we distinguish the following:

1) The property of normability, 2) The property of monotonity, 3) The property of \sigma-additivity, 4) The property of compactificability, 5) The logarithmic property, 6) The property of additivity, 7) The property of decomposability, 8) The G_\delta-enlargability property.

However, in the class T of all completely regular spaces (even in the class T_N of all normal spaces) none of these functions has the properties 1)-8) simultaneously.

In this connection the following questions appear naturally:

(\alpha) Is it true, that any (topologically closed) class of spaces, for which the functions ind, Ind and dim coincide and the common value of these functions has the properties 1)-8) simultaneously, is the subclass of the class T_SM?

(\beta) Do there exist integer-valued topologically invariant functions on the class T_SM different from d and having properties 1)-8) simultaneously?

(\gamma) Do there exist integer-valued topologically invariant functions defined on the classes of spaces wider than the class T_SM and having properties 1)-8) simultaneously? In particular, does a function of such type exist on the class T? If not, then what one can say about the existence of integer-valued topologically invariant functions defined on the given (wider then T_SM) class of spaces (in particular, on the class T) and having simultaneously properties from any preassigned subsystem of the system 1)-8)?

In the report several results in these directions obtained by the authors will be given.

Date received: June 5, 2003