### Abstract

By X(n), n≤1, we denote the n-th symmetric hyperspace of a metric space X as the space of non-empty finite subsets of X with at most n elements endowed with the Hausdorff metric. In this paper we shall describe the n-th symmetric hyperspace S^{1}(n) as a compactification of an open cone over σD^{n-2}, here D^{n-2} is the higher-dimensional dunce hat introduced by Andersen, Marjanović and Schori (1993) [2] if n is even, and D^{n-2} has the homotopy type of S^{n-2} if n is odd (see Andersen et al. (1993) [2]). Then we can determine the homotopy type of S^{1}(n) and detect several topological properties of S^{1}(n).

Original language | English |
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Pages (from-to) | 2613-2621 |

Number of pages | 9 |

Journal | Topology and its Applications |

Volume | 157 |

Issue number | 17 |

DOIs | |

Publication status | Published - 2010 Nov 1 |

Externally published | Yes |

### Keywords

- Compactification
- Dunce hat
- Symmetric hyperspace
- Symmetric product

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

Chinen, N., & Koyama, A. (2010). On the symmetric hyperspace of the circle.

*Topology and its Applications*,*157*(17), 2613-2621. https://doi.org/10.1016/j.topol.2010.07.012