Characterizing chainable, tree-like, and circle-like continua

Taras Banakh¹ ; Zdzisław Kosztołowicz² ; Sławomir Turek³

¹ Instytut Matematyki Uniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego Świętokrzyska 15 25-406 Kielce, Poland and Department of Mathematics Ivan Franko National University of Lviv Lviv, Ukraine
² Instytut Matematyki Uniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego Świętokrzyska 15 25-406 Kielce, Poland
³ Instytut Matematyki Uniwersytet Humanistyczno-Przyrodniczy Jana Kochanowskiego Świętokrzyska 15

Colloquium Mathematicum, Tome 124 (2011) no. 1, pp. 1-13.

Résumé

We prove that a continuum $X$ is tree-like (resp. circle-like, chainable) if and only if for each open cover $\mathcal U_4=\{U_1,U_2,U_3,U_4\}$ of $X$ there is a $\mathcal U_4$-map $f\colon X\to Y$ onto a tree (resp. onto the circle, onto the interval). A continuum $X$ is an acyclic curve if and only if for each open cover $\mathcal U_3=\{U_1,U_2,U_3\}$ of $X$ there is a $\mathcal U_3$-map $f\colon X\to Y$ onto a tree (or the interval $[0,1]$).

Détail

DOI : 10.4064/cm124-1-1

Keywords: prove continuum tree like resp circle like chainable only each cover mathcal there mathcal map colon tree resp circle interval continuum acyclic curve only each cover mathcal there mathcal map colon tree interval