DMS Continuum Theory Seminar

Speaker: David Lipham

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).