View source for Cube-like continuum

''<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c0272302.png" />-cube-like continuum''

A compactum (metrizable compactum) admitting, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c0272303.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c0272304.png" />-mapping onto the ordinary cube <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c0272305.png" />. If a compactum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c0272306.png" /> is the limit of a countable spectrum of compacta imbeddable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c0272307.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c0272308.png" /> is a subset of a cube-like continuum. The class of cube-like continua contains a universal element, i.e. a cube-like continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c0272309.png" /> such that every cube-like continuum is homeomorphic to some subspace of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c02723010.png" />.

====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>



====Comments====
In the special case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c02723011.png" />, these continua are also called snake-like, see [[#References|[a1]]].

In [[#References|[1]]] it is shown that a space is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c02723012.png" />-like if and only if it is homeomorphic to the limit of an inverse sequence of copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c027/c027230/c02723013.png" /> with surjective bounding mappings.

====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>