Irreducibility of product spaces with finitely many points removed

We prove an induction step that can be used to show that in certain cases the removal of finitely many points from a product space produces an irreducible space. For example, we show that whenever γ is less than אω, removing finitely many points from the product of γ many first countable compact spaces gives an irreducible space. This result answers questions asked privately by Alexander Arhangel’skii. If O is a collection of open sets corresponding to some topological space, we call O′ an open refinement of O if every member of O′ is an open set and a subset of a member of O. A collection S of sets is said to be minimal if for each S ∈ S there is an x ∈ S which is not in any other member of S. A topological space is said to be irreducible if every open cover of the space has a minimal open refinement covering the space (see [1, 2, 3, 4]). In this note we prove an induction step which can be used to show in certain cases that the removal of finitely many points from a product space yields an irreducible space. This induction step applies to regular limits, and the most immediate question left open is how to get past singular ones. One larger project, as we understand it, is to collect examples of noncompact irreducible spaces, though we cannot claim expertise on this topic. A cube in a product space X = ∏ a∈A Xa is a set of the form I f = {x ∈ X | ∀a ∈ dom(f) x(a) ∈ f(a)} for some finite function f with domain contained in A such that each f(a) is an open subset of Xa. We call the domain of f the support of I f . We say that a subspace Y of a product space X is cube-irreducible if every cover of Y by open subsets of X has a minimal open refinement covering Y and consisting of cubes. Note that every compact subspace of a product space is cube-irreducible. Following [5], we say that a topological space X is (γ,∞)-compact if every open cover of X has a subcover of cardinality less than γ. ∗The work in this paper was supported in part by NSF Grant DMS-0401603. The author would also like to thank Alexander Arhangel’skii, Dennis Burke and Sheldon Davis for introducing him to this topic.