DMS Continuum Theory Seminar

Speaker: David Lipham will continue speaking on last week's topic.

Abstract.  I plan to talk about "Singularities of meager composants and filament composants" in metric continua.  Given a continuum \(Y\) and a point \(x\) in \(Y\), 

• ​the meager composant of \(x\) in \(Y\) is the union of all nowhere dense subcontinua \(Y\) containing \(x\);
• the filament composant of \(x\) in \(Y\) is the union of all filament subcontinua of \(Y\) containing \(x\) (a subcontinuum \(L\) is filament if there is a neighborhood of \(L\) in which the component of \(L\) is nowhere dense); and
• a meager/filament composant \(P\) is said to be singular if there exists \(y\) in \(Y-P\) such that every connected subset of \(P\) limiting to \(y\) has closure equal to \(P\) (\(y\) is called a singularity of \(P\)).

To avoid trivial singularities, I will usually ​assume \(P\) is dense in \(Y\). 

I will prove that each singular dense meager composant of a continuum \(Y\) is homeomorphic to a traditional composant of an indecomposable continuum, even though \(Y\) may be decomposable.  If \(Y\) is homogeneous and has singular dense meager or filament composants, then I conjecture \(Y\) must be indecomposable (based on some partial results in this direction).