Positional dimension-like functions of the type Ind (Q719742)

In his master thesis [Subspace-dimension with respect to total spaces (1998)], \textit{R. Koga} gave a positional dimension-like function of the type Ind. Earlier papers of the authors and V. V. Tkachuk studied universalitity and dimensional properties of a positional dimension-like function of the type ind. In this paper it is shown that positional dimensions of type Ind can be stated in several manners. Let \(Q\) be a subset of \(X\). A family \({\mathcal B}\) of open subets of \(X\) is said to be a \(p(0)\)-big base (pos\((0)\)-big base) for \(Q\) in \(X\) if for every pair \((F,U)\) of subsets of \(X\), where \(F\) is closed in \(X\) and \(U\) is open in \(X\) and \(F\subset Q\cap U\), there exists a \(V\in{\mathcal B}\) such that \(F\subset Q\cap V\subset Q\cap U\) \((F\subset V\subset U)\). When we replace \(F\) closed in \(X\) by \(F^Q\) closed in \(Q\) in the above definition we obtain the definitions of a \(p(1)\)-big base (pos\((1)\)-big base) for \(Q\) in \(X\). We define the positional dimensions of type Ind as follows having ordinal values: \(p_0(0)\)-\(\text{Ind}(Q,X)= -1\) iff \(Q= X=\varnothing\), \(p_0(0)\)-\(\text{Ind}(Q,X)\leq\alpha\) iff there exists a \(p(0)\)-big base \({\mathcal B}\) for \(Q\) in \(X\) such that for every \(U\in{\mathcal B}\) we have \(p_0(0)\)-\(\text{Ind}(Q\cap \text{Bd}_X(U),\text{Bd}_X(U))<\alpha\). \(p_1(0)\)-\(\text{Ind}(Q,X)=-1\) iff \(Q=\varnothing\), \(p_1(0)\)-\(\text{Ind}(Q,X)\leq\alpha\) iff there exists a \(p(0)\)-big base \({\mathcal B}\) for \(Q\) in \(X\) such that for every \(U\in{\mathcal B}\) we have \(p_1(0)\)-\(\text{Ind}(Q\cap\text{Bd}_X(U), X)<\alpha\). \(\text{pos}_0(0)\)-\(\text{Ind}(Q,X)\), \(\text{pos}_1(0)\)-\(\text{Ind}(Q,X)\) are defined by using pos\((0)\)-big bases in the above definitions. Using \(p(1)\)-big bases and pos\((1)\)-big bases we again define four dimension functions, \(p_0(1)\)-\(\text{Ind}(Q,X)\), \(p_1(1)\)-\(\text{Ind}(Q,X)\), \(\text{pos}_0(1)\)-\(\text{Ind}(Q,X)\) and \(\text{pos}_1(1)\)-\(\text{Ind}(Q,X)\). Between these functions in general the following relations hold: \(p_i(0)\)-\(\text{Ind}(Q,X)\leq p_i(1)\)-\(\text{Ind}(Q,X)\) and \(\text{pos}_i(0)\)-\(\text{Ind}(Q,X)\leq\text{pos}_i(1)\)-\(\text{Ind}(Q,X)\) for \(i\in\{0,1\}\). Moreover, \(\text{Ind}(Q)\leq p_i(1)\)-\(\text{Ind}(Q,X)\) and also \(p_i(j)\)-\(\text{Ind}(Q, X)\leq\text{pos}_i(j)\)-\(\text{Ind}(Q,X)\) for \(i,j\in\{0,1\}\). By examples most of these inequalities are not to be replaced by equalities. For hereditarily normal spaces we have \(\text{Ind}(Q)= p_1(1)\)-\(\text{Ind}(Q,X)=\text{pos}_1(1)\)-\(\text{Ind}(Q,X)\). For \(T_1\)-spaces we have \(p_i\)-\(\text{Ind}(Q,X)\leq p_i(0)\)-\(\text{Ind}(Q,X)\) and \(\text{pos}_i\)-\(\text{Ind}(Q,X)\leq\text{pos}_i(0)\)-\(\text{Ind}(Q,X)\) for \(i\in\{0,1\}\), where \(p_i\)-Ind and \(\text{pos}_i\)-Ind are of type ind. Let in general \(p\)-\(\text{Ind}(Q,X)\) be one of the above defined dimension functions. The subspace theorem holds for subspaces \(K\subset Q: p\)-\(\text{Ind}(K,X)\leq p\)-\(\text{Ind}(Q,X)\). For a closed subspace \(Y\) of \(X\) such that \(Q\subset Y\subset X\) we have \(p\)-\(\text{Ind}(Q,Y)\leq p\)-\(\text{Ind}(Q,X)\). A partition theorem also holds for these dimension functions. In the case of \(\text{pos}_i(0)\)-\(\text{Ind}(Q,X)\) we also have a reverse statement for normal spaces. For the sum theorem in the case that \(Q_1\) is a closed subset of a hereditarily normal space \(X\) for \(p_1(i)\)-Ind and \(\text{pos}_1(i)\)-Ind we have that \(p\)-Ind \((Q_1\cup Q_2,X)\leq p\)-\(\text{Ind}(Q_1, X)\oplus p\)-\(\text{Ind}(Q_2,X)+ 1\). We have a product theorem for \(p_1(0)\)-Ind and \(\text{pos}_1(0)\)-Ind under the condition of compactness of \(X\) and \(Y\) and the finite sum holds in the product space \(X\times Y\). For subsets \(Q^X\subset X\) and \(Q^Y\subset Y\) we have that \(p\)-Ind \((Q^X\times Q^Y, X\times Y)\leq p\)-\(\text{Ind}(Q^X,X)\oplus p\)-\(\text{Ind}(Q^Y,Y)\). It is an open question to prove these sum- and product theorems for the other positional dimensions defined here. Some other question remain open. Can we find examples such that \(p_1(0)\)-\(\text{Ind}(Q,X)< \text{pos}_1(0)\)-\(\text{Ind}(Q,X)\)? Is there an example such that \(p_1(1)\)-\(\text{Ind}(Q,X)< \text{pos}_1(1)\)-\(\text{Ind}(Q,X)\)? Do we have for the metrizable case \(\text{pos}_1\)-\(\text{Ind}(Q,X)= \text{pos}_1(0)\)-\(\text{Ind}(Q,X)\)?