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 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).
| 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 |
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Keywords
- Compactification
- Dunce hat
- Symmetric hyperspace
- Symmetric product
ASJC Scopus subject areas
- Geometry and Topology
Cite this
On the symmetric hyperspace of the circle. / Chinen, Naotsugu; Koyama, Akira.
In: Topology and its Applications, Vol. 157, No. 17, 11.2010, p. 2613-2621.Research output: Contribution to journal › Article
}
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 -