Well-filterifications

Xiaodong Jia once asked the following question: is every core-compact, well-filtered space automatically locally compact? The question was solved positively this year by J. Lawson and X. Xi. I originally planned to try and explain their result. Even more recently, X. Xu, Ch. Shen, X. Xi and D. Zhao found a simpler solution, and I have changed my plans. My new plan for this time, and next time, is to explain what they have done. This time, we will concentrate on well-filterifications of topological space, which are just like sobrifications except ‘sober’ is replaced by ‘well-filtered T₀‘. Building one is not completely obvious. Xu, Shen, Xi and Zhao show that the will-filterification of X can be defined as its set of closed WD sets, a new notion that is intermediate between directed sets and irreducible sets; the proof also relies on a refinement of R. Heckmann and K. Keimel’s topological version of Rudin’s Lemma, which is interesting in its own right. Read the full post.