Topological vector space derived from a (Tallini) topological hypervector space

Subject Areas : General algebraic systems

R. Ameri ¹ (School of Mathematics, Statistics and Computer Sciences, University of Tehran, Tehran, Iran)
M. Hamidi ² (Department of Mathematics, Payame Noor University, Tehran, Iran)
A. Samadifam ³ (Department of Mathematics, Payame Noor University, Tehran, Iran)

Received: 2023-11-03 Accepted : 2024-06-07 Published : 2024-06-12

Keywords: Topological hypervector space, fundamental relation, complete part, homeomorphic,

Abstract :

In this paper, we consider a hypervector space (in the sense of Tallini) $V$ over a field $K$. We use the fundamental relation $\varepsilon^*$ over $V$, as the smallest equivalence relation on $V$, to derived the fundamental vector space ${V}/\varepsilon^*$. In this regards, we prove that if $V$ is a (resp. quasi) topological hypervector space, then the fundamental vector space ${V}/\varepsilon^*$ with the property that each open subset of it is a complete part, then its fundamental vector space ${V}/\varepsilon^*$ is a topological vector space. Finally, we prove that for a topological vector space $(V,+,\cdot,K)$ and every subspace $W$ of $V$, the hypervector space $(\overline{V},+,\circ,K)$ is a topological hypervector space and we will prove $\overline{V}/\varepsilon^*$ and $V/W$ are homeomorphic, where $\overline{V}=V$.

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