Geodesic
Characterizing chainable, tree-like, and circle-like continua
Taras Banakh 1   ; Zdzisław Kosztołowicz 2   ; Sławomir Turek 3
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 Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences

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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 $[0,1]$).
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

Taras Banakh  1   ; Zdzisław Kosztołowicz  2   ; Sławomir Turek  3

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 
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Taras Banakh; Zdzisław Kosztołowicz; Sławomir Turek. Characterizing chainable, tree-like, and circle-like continua. Colloquium Mathematicum, Tome 124 (2011) no. 1, pp. 1-13. doi: 10.4064/cm124-1-1

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