Difference between revisions of "Chainable continuum"

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(Created page with "A continuum $X$ is ''chainable'' if for each $\varepsilon>0$ there exists a map $f:X\to[0,1]$ such that diam$f^{-1}(t)<\varepsilon$ for each $t\in[0,1]$. ==Equivalent def...")
 
 
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A [[continuum]] $X$ is ''chainable'' if for each $\varepsilon>0$ there exists a map $f:X\to[0,1]$ such that diam$f^{-1}(t)<\varepsilon$ for each $t\in[0,1]$.
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A [[continuum]] $X$ is ''chainable'' if for each $\varepsilon>0$ there exists a map $f:X\to[0,1]$ such that  
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$$\mathrm{diam}(f^{-1}(t))<\varepsilon$$
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for each $t\in[0,1]$.
  
 
==Equivalent definitions==
 
==Equivalent definitions==

Latest revision as of 16:17, 27 August 2014

A continuum $X$ is chainable if for each $\varepsilon>0$ there exists a map $f:X\to[0,1]$ such that $$\mathrm{diam}(f^{-1}(t))<\varepsilon$$ for each $t\in[0,1]$.

Equivalent definitions

  • A continuum is chainable if for each open cover $\{U_i:i\in I\}$ of $X$, there exists a finite open cover $\{V_1,V_2,\dots,V_n\}$ of $X$ such that $V_i\cap V_j\neq\emptyset$ if and only if $|i-j|=0$ or $1$, and each $V_i$ is a subset of some $U_j$.

Examples of chainable continua

  1. Arc
  2. Topologist's sine curve
  3. Pseudo-arc