# Cactoid

From Encyclopedia of Mathematics

A locally connected continuum $C$ that is the closure of the sum of at most a countable number of spheres $S_i$ and simple arcs $D_i$ located in Euclidean space $\mathbf{E}^3$, and such that for each closed contour $L \subset C$ there exists exactly one sphere $S_i$ containing it. Cactoids, and they alone, are monotone images of the $2$-dimensional sphere $S^2$; also, every cactoid is a monotone open image of $S^2$.

**How to Cite This Entry:**

Cactoid.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Cactoid&oldid=34187

This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article