ИСТИНА

The talk presents an approach how to use dimension-like invariants in order to calculate the classical dimensions. B.Pasynkov [1969] established the finite-dimensionality of the product of two finite-dimensional compacta. V.Filippov [1972] constructed an example of compact spaces X, Y such that Ind X=1, Ind Y=2 and Ind XxY>3. We shall show that Ind XxY=4 in this case. The proof uses estimates of dimension Ind by dimension I introduced by S.Iliadis [2005] and the product theorem for I which is motivated by I.Lifanov's product theorem [1968]: Ind (X_1 x ... x X_k)<= Ind X_1+ ...+ Ind} X_k, if the weak form of the finite sum theorem is fulfilled in factors. The following natural question arises: whether every finite-dimensional in sense of Ind compacta has a normal base with respect to dimension I by which it is finite-dimensional and the finite sum theorem for I is fulfilled?