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

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

2010-03-28

arxiv.org/abs/1003.5341v2

Colloq. Math. 124 (2011), 1-13

Mathematics

General Topology

8 pages

Scientific paper

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

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