Geometry and Topology Weekly Seminar سمینار هفتگی هندسه و توپولوژی

TITLE
Connectifying a Topological Space by Adding one Point

SPEAKER
 Mohammad Reza Koushesh Isfahan University of Technology

TIME
 Wednesday, April 18, 2018, 15:30 - 17:00

VENUE   Lecture Hall 1, Niavaran Bldg.

SUMMARY

 In 1924, P. Alexandroff proved that a locally compact $T_2$-space has a $T_2$ one-point compactification (obtained by adding a point at infinity'') if and only if it is non-compact. He also asked for characterizations of spaces which have one-point connectifications. Here, we consider one-point connectifications, and in analogy with Alexandroff's theorem, we show that in the realm of $T_i$-spaces ($i=3\frac{1}{2},4,5$) a locally connected space has a one-point connectification if and only if it has no compact component. We explain how this theorem may be extended to the case $i=2$ and to the case $i=6$ under the set-theoretic assumption $\mathbf{MA}+\neg\mathbf{CH}$. Contrary to the case of the one-point compactification, a one-point connectification, if exists, may not be unique. We consider the collection of all one-point connectifications of a locally connected locally compact space in the realm of $T_i$-spaces ($i=3\frac{1}{2},4,5$). This collection, naturally partially ordered, is a compact conditionally complete lattice whose order-structure is rich enough to determine the topology of all Stone-Čech remainders of components of the space.
 تهران، ضلع‌ جنوبی ميدان شهيد باهنر (نياوران)، پژوهشگاه دانش‌های بنيادی، پژوهشکده رياضيات School of Mathematics, Institute for Research in Fundamental Sciences (IPM), Niavaran Bldg., Niavaran Square, Tehran ipmmath@ipm.ir   ♦   +98 21 22290928   ♦  math.ipm.ir