Chainable continuum

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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 $$\mathrm{diam}(f^{-1}(t))<\varepsilon$$ 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