DMS Set Theoretic Topology Seminar

Speaker: Vladimir Tkachuk

Title: Every Lindelof Sigma-space has the van Douwen property D.

Abstract: After completing the proof of Hodel’s theorem, we will present the proof of Buzyakova’s theorem which states that every Lindelof Sigma-space is a \(D\)-space in the sense of van Douwen. Recall that \(N\) is a neighborhood assignment in a space \(X\) if \(N\) maps \(X\) to the topology of \(X\) and \(N(x)\) contains \(x\) for any \(x\) in \(X\). It is said that \(X\) is a \(D\)-space if for any neighborhood assignment \(N\) on \(X\), there exists a closed discrete set \(D \subset X\) such that \(\bigcup\{N(x): x\in D\} =X\). It is an old unsolved problem of van Douwen whether every Lindelof space has the D-property and it is not trivial at all to prove that each Lindelof Sigma-space must be a \(D\)-space.