Cube-like continuum
From Encyclopedia of Mathematics
-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=25096
Cube-like continuum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cube-like_continuum&oldid=25096
This article was adapted from an original article by L.G. Zambakhidze (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article