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Difference between revisions of "Cube-like continuum"

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$n$-cube-like continuum''
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''$n$-cube-like continuum''
  
 
A compactum (metrizable compactum) admitting, for any $\epsilon > 0$, an $\epsilon$-mapping onto the ordinary cube $I^n$. If a compactum $X$ is the limit of a countable spectrum of compacta imbeddable in $I^n$, then $X$ is a subset of a cube-like continuum. The class of cube-like continua contains a universal element, i.e. a cube-like continuum $U$ such that every cube-like continuum is homeomorphic to some subspace of $U$.
 
A compactum (metrizable compactum) admitting, for any $\epsilon > 0$, an $\epsilon$-mapping onto the ordinary cube $I^n$. If a compactum $X$ is the limit of a countable spectrum of compacta imbeddable in $I^n$, then $X$ is a subset of a cube-like continuum. The class of cube-like continua contains a universal element, i.e. a cube-like continuum $U$ such that every cube-like continuum is homeomorphic to some subspace of $U$.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> B.A. Pasynkov,  "On universal compacta"  ''Russian Math. Surveys'' , '''21''' :  4  (1966)  pp. 77–86  ''Uspekhi Mat. Nauk'' , '''21''' :  4  (1966)  pp. 91–100</TD></TR></table>
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|valign="top"|{{Ref|Pa}}||valign="top"| B.A. Pasynkov,  "On universal compacta"  ''Russian Math. Surveys'', '''21''' :  4  (1966)  pp. 77–86  ''Uspekhi Mat. Nauk'', '''21''' :  4  (1966)  pp. 91–100
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====Comments====
 
====Comments====
In the special case $n=1$, these continua are also called snake-like, see [[#References|[a1]]].
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In the special case $n=1$, these continua are also called snake-like, see {{Cite|Bi}}.
  
In [[#References|[1]]] it is shown that a space is $I^n$-like if and only if it is homeomorphic to the limit of an inverse sequence of copies of $I^n$ with surjective bounding mappings.
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In {{Cite|Pa}} it is shown that a space is $I^n$-like if and only if it is homeomorphic to the limit of an inverse sequence of copies of $I^n$ with surjective bounding mappings.
  
 
====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> R.H. Bing,  "Snake-like continua"  ''Duke Math. J.'' , '''18'''  (1951)  pp. 553–663</TD></TR></table>
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|valign="top"|{{Ref|Bi}}||valign="top"| R.H. Bing,  "Snake-like continua"  ''Duke Math. J.'', '''18'''  (1951)  pp. 553–663
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Latest revision as of 22:28, 22 April 2012


$n$-cube-like continuum

A compactum (metrizable compactum) admitting, for any $\epsilon > 0$, an $\epsilon$-mapping onto the ordinary cube $I^n$. If a compactum $X$ is the limit of a countable spectrum of compacta imbeddable in $I^n$, then $X$ is a subset of a cube-like continuum. The class of cube-like continua contains a universal element, i.e. a cube-like continuum $U$ such that every cube-like continuum is homeomorphic to some subspace of $U$.

References

[Pa] B.A. Pasynkov, "On universal compacta" Russian Math. Surveys, 21 : 4 (1966) pp. 77–86 Uspekhi Mat. Nauk, 21 : 4 (1966) pp. 91–100

Comments

In the special case $n=1$, these continua are also called snake-like, see [Bi].

In [Pa] it is shown that a space is $I^n$-like if and only if it is homeomorphic to the limit of an inverse sequence of copies of $I^n$ with surjective bounding mappings.

References

[Bi] R.H. Bing, "Snake-like continua" Duke Math. J., 18 (1951) pp. 553–663
How to Cite This Entry:
Cube-like continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cube-like_continuum&oldid=25093
This article was adapted from an original article by L.G. Zambakhidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article