Connectedness of suns in the space~$c_0$

A. R. Alimov

Izvestiya. Mathematics , Tome 69 (2005) no. 4, pp. 651-666

Résumé

We study the question of the connectedness of suns in the space $c_0$. We show that any sun $M$ in $c_0$ is m-connected (in the sense of Brown). It follows that $M$ is monotonically path-connected and the intersection of $M$ with an arbitrary ball in $c_0$ is monotonically path-connected (and, in particular, path-connected). On the other hand, we establish that every approximatively compact m-connected set in $c_0$ is a sun in $c_0$. For $X=c_0$, $c$ or $\ell^\infty$, it is proved that the intersection of a sun in $X$ with a finite-dimensional coordinate subspace $H_n\subset X$ is a $P$-acyclic sun in $H_n$.