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On relative dimensions. (Russian) Zbl 0625.54040

General topology, Function spaces and dimension, Work collect., Moskva 1985, 67-117 (1985).
[For the entire collection see Zbl 0594.00012.]
The theory of relative dimension functions d(X,Y), I(X,Y), i(X,Y) in the class of Hausdorff completely regular spaces is discussed. Some results are known [Soobshch. Akad. Nauk Gruz. SSR 87, 289-292 (1977; Zbl 0371.54052) and ibid. 89, 277-280 (1978; Zbl 0416.54032)] but the proofs are given here. Other results with proofs are pulished for the first time. If X is a subset of Y then \(by\quad Z(X,Y)\) the trace of the family of zero-sets Z(X) on the set X is denoted. The dimension d(X,Y) of a space \(X\subset Y\) in relation to Y is not greater than n, d(X,Y)\(\leq n\), if for each finite \({\mathcal C}Z(X,Y)\)-covering \(\Omega\) of the space X there is a finite refinement \(\omega\) such that ord \(\omega\leq n+1\). Some properties of the function d are as follows.
If \(X\subset Y\) then \(d(X,Y)=d(X,\omega (Z(X,Y)))=\dim \omega (Z(X,Y))\). Let \(X\subset Y\), \(X=\cup^{\infty}_{i=1}Z_ i\), \(Z_ i\in Z(X,Y)\), \(d(Z_ i,Y)\leq n\); then d(X,Y)\(\leq n\). If A,B\(\subset X\), then \(d(A\cup B,X)\leq d(A,X)+d(B,X)+1\). If \(X\subset Y\subset T\) then d(X,T)\(\leq d(Y,T)\). Dimension d(X,Y) is the least integer \(n\geq 0\) such that for an arbitrary \({\mathcal C}Z(X,Y)\)-covering (finite) \(\omega\) of X there exists an \(\omega\)-mapping from X into an n-dimensional polyhedron.
For \(X\subset Y\) relative inductive dimensions I(X,Y) and i(X,Y) are defined. \(Ind_ 0\) and \(ind_ 0\) are defined as \(Ind_ 0X=I(X,X)\), \(ind_ 0X=i(X,X)\). Some properties of these functions are proved. There are addition theorems, subset theorems, sum theorems, product theorems and some other properties. The relations between several dimension functions are given \((i(X,Y)\leq I(X,Y),I(X,Y)\leq Ind_ 0Y\); \(ind_ 0X\leq Ind_ 0X\), d(X,Y)\(\leq I(X,Y)\), dim \(X\leq Ind_ 0X\) and others).
Dimension of perfectly \(\kappa\)-normal spaces and infinite dimensional spaces are also studied.
Reviewer: D.Adnajević

MSC:

54F45 Dimension theory in general topology
54Bxx Basic constructions in general topology