Difference between revisions of "Cube-like continuum"
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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==== | ||
| − | + | {| | |
| − | + | |- | |
| − | + | |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 | |
| + | |- | ||
| + | |} | ||
====Comments==== | ====Comments==== | ||
| − | In the special case $n=1$, these continua are also called snake-like, see | + | In the special case $n=1$, these continua are also called snake-like, see {{Cite|Bi}}. |
| − | In | + | 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==== | ||
| − | + | {| | |
| + | |- | ||
| + | |valign="top"|{{Ref|Bi}}||valign="top"| R.H. Bing, "Snake-like continua" ''Duke Math. J.'', '''18''' (1951) pp. 553–663 | ||
| + | |- | ||
| + | |} | ||
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
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