Some properties of topological hedgehogs

D. Yu. Lyakhovets ; A. V. Osipov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 88 (2024), pp. 37-52

Résumé

The topological spaces called Euclidean hedgehogs are considered. These are subspaces of the Euclidean spaces $\mathbf{R^n}$ with the following property: together with each of their points, they contain the entire segment connecting the given point with the point of origin. It is proved that for all $n\geqslant 2$ there exist pairwise non-homeomorphic Euclidean hedgehogs in $\mathbf{R^n}$. It is also proved that for every countable Euclidean hedgehog there exists a flat hedgehog homeomorphic to it. We also consider two topological spaces: the quasimetric hedgehog and the quotient hedgehog, which have the following cardinal and hereditary invariants: weight, character, density, spread, extent, cellularity, tightness, number of open sets, and Lindelof number. Finally, sequential hedgehogs are considered that are topologically embedded in function spaces. Criteria are given for the topological embedding of sequential hedgehogs in the space of continuous functions and in the space of Baire functions.

Détail

Keywords: Euclidean hedgehog, quasi-metric, factor topology, Sorgenfrey line, metric hedgehog, sequential hedgehog, space of continuous functions, space of Baire functions, topological embedding.
Mots-clés : cardinal invariants