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

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 $\overline{V}=V$.