the nearly open set A topological space on $X$ ‎denotes  $‎(X;‎\tau‎  )‎‎$‎‎  . For a subset ‎‎$‎S‎$‎ of$(‎X;‎\tau‎)‎‎$‎‎‎,‎‎$‎int‎_{‎\tau‎}‎ (S)‎‎$‎‎‎ and $‎cl‎_{‎\tau‎}‎ (S)‎‎$‎‎  stand for the interior of S and the closure of S, respectively. If ‎‎$ ‎Y ‎\subset ‎X‎$‎,then $‎(Y; ‎\tau‎_{‎Y}‎‎)$‎‎‎denotes a subspace of ‎‎$‎(X; ‎\tau‎)‎‎$‎, and $ ‎int‎_{‎\tau‎‎_{Y}‎}‎‎ $‎, $ ‎cl‎_{‎\tau‎‎_{Y}‎}‎‎ $‎‎‎ are interior and closure operators (respectively) in ‎‎$‎(Y; ‎\tau‎_{‎Y}‎‎)$‎‎‎.‎
‎A set S is said to be ‎ semi-preopen set  in ‎$‎(X;‎\tau‎  )‎‎$‎, if ‎‎$‎ S ‎\subset‎ cl‎_{‎\tau‎}‎ (int‎_{‎\tau‎}‎(‎‎cl‎_{‎\tau} ‎(S)‎)‎)‎‎$‎.‎
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Let ‎‎$‎A ‎\subset ‎X‎_{0}‎‎ ‎\subset X ‎‎‎‎$‎where ‎‎$‎X‎$ ‎is a‎
‎topological ‎space ‎and ‎‎$‎X_{0}‎$ ‎is a‎ ‎subspace. ‎If‎ ‎‎$‎A‎$
‎is  a ‎semi-preopen ‎set ‎in‎ $‎(X;‎\tau‎  )‎$, ‎is ‎semi-preopen ‎set
‎in‎ ‎‎$‎(X_0; ‎\tau‎‎_{‎X‎_{0}‎‎}‎ )‎‎$‎‎?‎‎


Put $B=\operatorname{int}_\tau \operatorname{cl}_\tau A$. Let $x$ be any point of $A$ and $O$ be any $\tau$-open neighborhood of $x$. Since $x\in A\subset \operatorname{cl}_\tau B$, a set $O\cap B$ is non-empty. Since $O\cap B$ is $\tau$-open and $A$ is $\tau$-dense in $O\cap B$,  there exists a point $y\in A\cap O\cap B$. Then $X_0\cap O\cap B$ is a
$\tau_{X_0}$-open neighborhood of $y$ contained in $X_0\cap\operatorname{cl}_\tau A=\operatorname{cl}_{\tau_{X_0}} A$.
That is $y\in \operatorname{int}_{\tau_{X_0}} \operatorname{cl}_{\tau_{X_0}} A$, which follows
$x\in \operatorname{cl}_{\tau_{X_0}} \operatorname{int}_{\tau_{X_0}} \operatorname{cl}_{\tau_{X_0}} A$, and so
$A\subset \operatorname{cl}_{\tau_{X_0}} \operatorname{int}_{\tau_{X_0}} \operatorname{cl}_{\tau_{X_0}} A$.
