Cube-like continuum

$n$-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