Order-theoretic properties of bases in topological spaces I

Mathematics – General Topology

Scientific paper

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20 pages

Scientific paper

We study some cardinal invariants of an order-theoretic fashion on products and box products of topological spaces. In particular, we concentrate on the Noetherian type (Nt), defined by Peregudov in the 1990s. Some highlights of our results include: 1) There are spaces $X$ and $Y$ such that $Nt(X \times Y) < \min\{Nt(X), Nt(Y)\}$. 2) In several classes of compact spaces, the Noetherian type is preserved by their square and their dense subspaces. 3) The Noetherian type of some countably supported box products cannot be determined in ZFC. In particular, it is sensitive to square principles and some Chang Conjecture variants. 4) PCF theory can be used to provide ZFC upper bounds to Noetherian type on countably supported box products. The underlying combinatorial notion is a weakening of Shelah's freeness.

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