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Algebraic characterisation of hyperspace corresponding to topological vector space | ||
| Journal of Hyperstructures | ||
| دوره 11، شماره 1، شهریور 2022، صفحه 48-64 اصل مقاله (343.79 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22098/jhs.2023.2527 | ||
| نویسندگان | ||
| Jayeeta Saha* 1؛ Sandip Jana2 | ||
| 1Department of Mathematics, Vivekananda College,Thakurpukur, Kolkata, West Bengal, India | ||
| 2Department of Pure Mathematics, University of Calcutta, Kolkata, West Bengal, India | ||
| چکیده | ||
| Let X be a Hausdor topological vector space over the field of real or complex numbers. When Vietoris topology is given, the hyperspace ℘(X) of all nonempty compact subsets of X forms a topological exponential vector space over the same field. Exponential vector space [shortly, evs] is an algebraic ordered extension of vector space in the sense that every evs contains a vector space, and conversely, every vector space can be embedded into such a structure. A semigroup structure, a scalar multiplication and a partial order with some compatible topology comprise the topological evs structure. In this study, we have shown that besides ℘(X), there are other hyperspaces namely P(X), PBal(X) PCV (X), PNθ (X), PS(X), Pθ(X) which have the same structure. To characterise the hyperspaces P(X), ℘(X) in light of evs, we have introduced some properties of evs which remain invariant under order-isomorphism. We have also introduced the concept of primitive function of an evs, which plays an important role in such characterisation. Lastly, with the help of these properties, we have characterised ℘(X) as well as P(X) as exponential vector spaces. | ||
| کلیدواژهها | ||
| Exponential vector space؛ topological exponential vector space؛ hyperspaces؛ order-isomorphism؛ primitive function | ||
| مراجع | ||
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[1] E. Michael; Topologies on spaces of subsets, Trans.Amer.Math.Soc. 71(1951),152-182. [2] Leopoldo Nachbin, Topology And Order, D.Van Nostrand Company, Inc. (1965) [3] Priti Sharma, Sandip Jana, An algebraic ordered extension of vector space, Transactions of A. Razmadze Mathematical Institute; 172 (2018) 545-558, Elsevier; https ://doi.org/10.1016/j.trmi.2018.02.002. [4] S. Ganguly, S. Mitra, S. Jana, An Associated Structure Of A Topological Vector Space, Bull. Cal. Math. Soc; Vol-96, No.6 (2004), 489-498. [5] S. Ganguly, S. Mitra, More on topological quasi-vector space, Revista de la Academia Canaria de Ciencias; Vol.22 No.1-2 (2010), 45-58. [6] S. Jana, J. Saha, A Study of Topological quasi-vector Spaces, Revista de la Academia Canaria de Ciencias; XXIV, No.1 (2012), 7-23. | ||
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