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Difference between revisions of "Cactoid"

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A locally connected continuum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020040/c0200401.png" /> that is the closure of the sum of at most a countable number of spheres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020040/c0200402.png" /> and simple arcs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020040/c0200403.png" /> located in Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020040/c0200404.png" />, and such that for each closed contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020040/c0200405.png" /> there exists exactly one sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020040/c0200406.png" /> containing it. Cactoids, and they alone, are monotone images of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020040/c0200407.png" />-dimensional sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020040/c0200408.png" />; also, every cactoid is a monotone open image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c020/c020040/c0200409.png" />.
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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 $S^2$.
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Revision as of 22:19, 1 November 2014

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=12209
This article was adapted from an original article by B.A. Efimov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article