Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces
Izvestiya. Mathematics , Tome 78 (2014) no. 4, pp. 641-655
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We prove that every boundedly compact $\operatorname{m}$-connected
(Menger-connected) set is monotone path-connected and is a sun
in a broad class of Banach spaces (in particular, in separable spaces).
We show that the intersection of a boundedly compact monotone
path-connected ($\operatorname{m}$-connected) set with
a closed ball is cell-like (of trivial shape) and, in particular,
acyclic (contractible in the finite-dimensional case) and is a sun.
We also prove that every boundedly weakly compact
$\operatorname{m}$-connected set is monotone path-connected.
In passing, we extend the Rainwater–Simons weak convergence
theorem to the case of convergence with respect to the associated norm
(in the sense of Brown).
Keywords:
sun, acyclic set, cell-like set, monotone path-connected set, Menger connectedness,
$d$-convexity, Menger convexity, Rainwater–Simons theorem.
@article{IM2_2014_78_4_a0,
author = {A. R. Alimov},
title = {Monotone path-connectedness and solarity of {Menger-connected} sets in {Banach} spaces},
journal = {Izvestiya. Mathematics },
pages = {641--655},
publisher = {mathdoc},
volume = {78},
number = {4},
year = {2014},
language = {en},
url = {http://geodesic.mathdoc.fr/item/IM2_2014_78_4_a0/}
}
A. R. Alimov. Monotone path-connectedness and solarity of Menger-connected sets in Banach spaces. Izvestiya. Mathematics , Tome 78 (2014) no. 4, pp. 641-655. http://geodesic.mathdoc.fr/item/IM2_2014_78_4_a0/