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.