topological vector space derived from a (tallini) topological hypervector space

N 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 $overline{v}=v$.