View source for Snake-like continuum

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A [[Continuum|continuum]] which, for any $\epsilon>0$, admits an open covering whose nerve (cf. [[Nerve of a family of sets|Nerve of a family of sets]]) is a finite linear complex. In other words, for any $\epsilon>0$ the continuum must be covered by a finite system $G_n$, $n=1,\ldots,p$, of open sets such that the diameter of each $G_n$ is less than $\epsilon$ and $G_i\cap G_j\neq\emptyset$ if and only if $|i-j|=1$ (such a system is called an $\epsilon$-chain). Every snake-like continuum is irreducible (see [[Irreducible continuum|Irreducible continuum]]) between any pair of its points. Every subcontinuum of a snake-like continuum is snake-like. Two hereditarily-indecomposable snake-like continua (see [[Indecomposable continuum|Indecomposable continuum]]) containing more than one point are homeomorphic; these are known as pseudo-arcs (cf. [[Pseudo-arc|Pseudo-arc]]). Every snake-like continuum is topologically imbeddable in the plane. Any homogeneous snake-like continuum is a pseudo-arc. Every snake-like continuum is the continuous image of a pseudo-arc and the limit of the inverse spectrum of arcs.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  K. Kuratowski,   "Topology" , '''2''' , Acad. Press  (1968)  (Translated from French)</TD></TR></table>



====Comments====
A snake-like continuum is also called a chainable continuum.