Tychonoff’s theorem for hyperspaces
HTML articles powered by AMS MathViewer
- by Frank A. Chimenti
- Proc. Amer. Math. Soc. 37 (1973), 281-286
- DOI: https://doi.org/10.1090/S0002-9939-1973-0307141-1
- PDF | Request permission
Abstract:
If $\exp ({X_i})\backslash \{ \emptyset \}$ is equipped with a topology that preserves the topological convergence of nets of sets for every $i \in I$, then the Tychonoff product of the family $\{ \exp ({X_i})\backslash \{ \emptyset \} :i \in I\}$ is compact if and only if ${X_i}$ is compact for every $i \in I$. A similar result concerning sequential compactness is valid, for countable I.References
- Garrett Birkhoff, Moore-Smith convergence in general topology, Ann. of Math. (2) 38 (1937), no. 1, 39–56. MR 1503323, DOI 10.2307/1968508
- D. Bushaw, Elements of general topology, John Wiley & Sons, Inc., New York-London, 1963. MR 159298
- F. A. Chimenti, On the sequential compactness of the space of subsets, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 20 (1972), 221–226 (English, with Russian summary). MR 310841
- J. M. G. Fell, A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Amer. Math. Soc. 13 (1962), 472–476. MR 139135, DOI 10.1090/S0002-9939-1962-0139135-6
- Zdeněk Frolík, Concerning topological convergence of sets, Czechoslovak Math. J. 10(85) (1960), 168–180 (English, with Russian summary). MR 116303, DOI 10.21136/CMJ.1960.100401 F. Hausdorff, Mengenlehre, 3rd ed., de Gruyter, Berlin, 1937; English transl., Chelsea, New York, 1962, p. 168. MR 25 #4999.
- James Keesling, On the equivalence of normality and compactness in hyperspaces, Pacific J. Math. 33 (1970), 657–667. MR 267516, DOI 10.2140/pjm.1970.33.657
- Y.-F. Lin, Tychonoff’s theorem for the space of multifunctions, Amer. Math. Monthly 74 (1967), 399–400. MR 210074, DOI 10.2307/2314568
- Ernest Michael, Topologies on spaces of subsets, Trans. Amer. Math. Soc. 71 (1951), 152–182. MR 42109, DOI 10.1090/S0002-9947-1951-0042109-4
- S. Mrówka, Some comments on the space of subsets, Set-Valued Mappings, Selections and Topological Properties of $2^X$ (Proc. Conf., SUNY, Buffalo, N.Y., 1969) Lecture Notes in Math., Vol. 171, Springer, Berlin-New York, 1970, pp. 59–63. MR 270327
Bibliographic Information
- © Copyright 1973 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 37 (1973), 281-286
- MSC: Primary 54B10; Secondary 54B20
- DOI: https://doi.org/10.1090/S0002-9939-1973-0307141-1
- MathSciNet review: 0307141