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Difference between revisions of "Snake-like continuum"

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A [[Continuum|continuum]] which, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085930/s0859301.png" />, 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085930/s0859302.png" /> the continuum must be covered by a finite system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085930/s0859303.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085930/s0859304.png" />, of open sets such that the diameter of each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085930/s0859305.png" /> is less than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085930/s0859306.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085930/s0859307.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085930/s0859308.png" /> (such a system is called an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s085/s085930/s08593010.png" />-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.
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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.
  
 
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Latest revision as of 12:29, 12 April 2014

A continuum which, for any $\epsilon>0$, admits an open covering whose nerve (cf. 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) 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) containing more than one point are homeomorphic; these are known as pseudo-arcs (cf. 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

[1] K. Kuratowski, "Topology" , 2 , Acad. Press (1968) (Translated from French)


Comments

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

How to Cite This Entry:
Snake-like continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Snake-like_continuum&oldid=11338
This article was adapted from an original article by A.A. Mal'tsev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article