A subset $A$ of a topological space $X$ is said to be semi-open if there exists an open set $O$ in $X$ such that $O \subset A \subset Cl(O)$, or equivalently if $A \subset Cl(Int(A))$. $SO(X)$ denotes the collection of all semi-open sets in $X$.
Recall that a set $U \subset X$ is a semi-open neighborhood of a point $x \in X$ if there exists $A \in SO(X)$ such that $x \in A \subset U$. A set $A \subset X$ is semi-open in $X$ if and only if $A$ is semi-open neighborhood of each of its points. If a semi-open neighborhood $U$ of a point $x$ is a semi-open set, we say that $U$ is a semi-open neighborhood of $x$.
Definition An s-topological vector space $(X , \tau )$ is a vector space $X$ over the field $\mathbb F$ ($\mathbb R$ or $\mathbb C$) with a topology $\tau$ defined on $X$ and standard topology on $\mathbb F$ such that:
1) For each $x, y \in X$, and for each open neighborhood $W$ of $x + y$ in $X$, there exist semi-open neighborhoods $U$ and $V$ of $x$ and $y$ respectively in $X$, such that $U + V \subseteq W$.
2) For each $\lambda \in \mathbb F$, $x \in X$ and for each open neighborhood $W$ of $\lambda x$ in $X$, there exist semi-open neighborhoods $U$ of $\lambda$ in $\mathbb F$ and $V$ of $x$ in $X$ such that $UV\subseteq W$.
I need to prove that the following example is an s-topological vector space
Example: Let $X = \mathbb R$ be a vector space of real numbers over the field $\mathbb F = \mathbb R$ and let $\tau$ be a topology on $X$ induced by open intervals $(a, b)$ and the sets $[1, c)$ where $a, b, c \in\mathbb R$.