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Remarks on best approximation in R-trees

William Kirk, Bancha Panyanak (2009)

Annales UMCS, Mathematica

An R-tree is a geodesic space for which there is a unique arc joining any two of its points, and this arc is a metric segment. If X is a closed convex subset of an R-tree Y, and if T: X → 2Y is a multivalued mapping, then a point z for which [...] is called a point of best approximation. It is shown here that if T is an ε-semicontinuous mapping whose values are nonempty closed convex subsets of Y, and if T has at least two distinct points of best approximation, then T must have a fixed point. We...

Remarks on cardinal inequalities in convergence spaces

Kazushi Yoshitomi (2021)

Mathematica Bohemica

We extend the Noble and Ulmer theorem and the Juhász and Hajnal theorems in set-theoretic topology. We show that a statement analogous to that in the former theorem is valid for a family of almost topological convergences, whereas statements analogous to those in the latter theorems hold for a pretopologically Hausdorff convergence.

Remarks on dense subspaces

Eva Murtinová (2004)

Czechoslovak Mathematical Journal

Some constructions of spaces with/without dense subspaces satisfying stronger separation axioms are presented.

Remarks on extremally disconnected semitopological groups

Igor V. Protasov (2002)

Commentationes Mathematicae Universitatis Carolinae

Answering recent question of A.V. Arhangel'skii we construct in ZFC an extremally disconnected semitopological group with continuous inverse having no open Abelian subgroups.

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