On spaces without non-trivial subcontinua and the dimension of their products
All spaces considered in this paper are separable and metrizable. For a space \(X\), \(X_{(0)}\) denotes the set of all points at which the dimension is \(0\). A space \(X\) is said to be \textsl{splintered} if every open cover of \(X\) has a countable refinement by mutually disjoint closed sets. Since no continuum can be written as the union of countably many non-empty disjoint closed sets, it follows that every compact subspace of a splintered space is zero-dimensional. A space is said to be \textsl{strongly splintered} if there exist closed sets \(F_i\) in \(X\) such that \(X=\bigcup_{i=1}^\infty(F_i)_{(0)}\). In Section 1 of this paper the relationship between these properties and almost zero-dimension\-al\-ity, weakly 1-dimensionality and the concept of an \(L\)-embedded subspace of a compact space are discussed. Section 2 contains preliminary results while the main theorem of Section 3 is a proof that a strongly splintered space is splintered. Section 4 contains examples of products of splintered and strongly splintered spaces while in Section 5 it is shown that for each \(\alpha<\omega_1\) there is a strongly splintered topologically complete space of small transfinite dimension \(\alpha\). The final section is dedicated to the following theorem of Tomaszewski: If \(X\) is weakly \(n\)-dimensional and \(Y\) is weakly \(m\)-dimensional, then \(\dim(X\times Y)\leq\dim X+\dim Y-1\). The authors show that the inductive argument in the proof originally given by Tomaszewski fails in case \(n=1\) and a new proof for this case is given here.
scientific article; zbMATH DE number 2117942
scientific article