### 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 |
---|---|

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 |

Externally published | Yes |

### Fingerprint

### Keywords

- Compactification
- Dunce hat
- Symmetric hyperspace
- Symmetric product

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

*Topology and its Applications*,

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

**On the symmetric hyperspace of the circle.** / Chinen, Naotsugu; Koyama, Akira.

Research output: Contribution to journal › Article

*Topology and its Applications*, vol. 157, no. 17, pp. 2613-2621. https://doi.org/10.1016/j.topol.2010.07.012

}

TY - JOUR

T1 - On the symmetric hyperspace of the circle

AU - Chinen, Naotsugu

AU - Koyama, Akira

PY - 2010/11

Y1 - 2010/11

N2 - 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 S1(n) as a compactification of an open cone over σDn-2, here Dn-2 is the higher-dimensional dunce hat introduced by Andersen, Marjanović and Schori (1993) [2] if n is even, and Dn-2 has the homotopy type of Sn-2 if n is odd (see Andersen et al. (1993) [2]). Then we can determine the homotopy type of S1(n) and detect several topological properties of S1(n).

AB - 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 S1(n) as a compactification of an open cone over σDn-2, here Dn-2 is the higher-dimensional dunce hat introduced by Andersen, Marjanović and Schori (1993) [2] if n is even, and Dn-2 has the homotopy type of Sn-2 if n is odd (see Andersen et al. (1993) [2]). Then we can determine the homotopy type of S1(n) and detect several topological properties of S1(n).

KW - Compactification

KW - Dunce hat

KW - Symmetric hyperspace

KW - Symmetric product

UR - http://www.scopus.com/inward/record.url?scp=77956898283&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=77956898283&partnerID=8YFLogxK

U2 - 10.1016/j.topol.2010.07.012

DO - 10.1016/j.topol.2010.07.012

M3 - Article

VL - 157

SP - 2613

EP - 2621

JO - Topology and its Applications

JF - Topology and its Applications

SN - 0166-8641

IS - 17

ER -