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On embeddability of cones in Euclidean spaces. (English) Zbl 0857.54015

Let \(K_1\), \(K_2\), \(K_3\), and \(K_4\), are the well-known Kuratowski curves. By proving that the cones over these graphs are not embeddable in \(\mathbb{R}^3\), the author proves that if \(X\) is a locally connected continuum and its cone is embeddable in \(\mathbb{R}^n\), for \(n\leq 3\), then \(X\) is embeddable in \(S^{n-1}\). Finally, using Blankinship’s wild arcs lying in the \(n\)-dimensional balls, he proves that for each \(n\geq3\) a locally connected continuum \(X_n\) exists, which is not embeddable in \(\mathbb{R}^n\) but its cone is embeddable in \(\mathbb{R}^{n+1}\). At the end of the paper, an interesting question is stated: If \(X\) is a locally connected continuum with its \(n\)th cone embeddable in \(\mathbb{R}^{n+2}\), is it true that \(X\) is embeddable in \(S^2\)?

MSC:

54C25 Embedding
54F15 Continua and generalizations
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