On dense subsets of Tychonoff products of \(T_{1}\)-spaces
It is well-known that the Tychonoff product of \(\mathfrak c\) many separable spaces is a separable space (by the classical Hewitt-Marczewski-Pondiczery Theorem). In the paper under review, the author considers the problem of the existence, in products of \(\mathfrak c\) separable spaces, of a dense countable subset with additional properties. The main result states as follows: let \(Z\) be a separable, infinite topological space, and let \(\eta\) be a family of non-empty, countable subsets of \(Z\), \(|\eta| \leqslant \mathfrak c\). Then there is a countable dense subset \(Q \subseteq Z^{\mathfrak c}=\prod\limits_{X \subseteq \omega} ZX\) satisfying the following condition: (*) every countable set \(E \subseteq Q\) contains a countable subset \(E' \subseteq E\) such that for every \(G \in \eta\) there is some \(X \subseteq \omega\) such that \(\prod_X(E') = G\) and \(\prod_{X|_{E'}}: E' \to G\) is a finite-to-one mapping. The author deduces a number of consequences of the above stated result. For instance, it is shown in the paper that if \(Z\) is a separable \(T_1\) space with more than one point then there is a countable dense set \(Q \subseteq Z^{\mathfrak c}\) which does not contain nontrivial convergent sequences in \(Z^{\mathfrak c}\). All proofs of the paper use the notion of independent matrices of subsets of a countable set.
scientific article; zbMATH DE number 6946127
scientific article