TITLE: On the hereditary properties of $\theta-$closed subsets
QUESTION [1 upvotes]: While working on $\theta-$closures and $\theta$-closed hulls, I came with a rather interesting question, but first some definitions:
Given a topological space $X$ and a set $A\subset X$ we say that $x\in X$ is a $\theta-$adherent point of $A$ if every open neighborhood $U$ of $x$ satisfies $\overline{U}\cap A\neq \emptyset$, and we define the $\theta-$closure of $A$ as the set of all $\theta-$adherent points of $A$. One must notice that the $\theta-$closure is not an idempotent operator in general, i.e. It is no true (In general) that $Cl_\theta (A)=Cl_\theta (Cl_\theta (A))$. We also say that a set $C$ is $\theta -$closed if $C=Cl_\theta (C)$, and now we can define the $\theta-$closed hull of a set $A$ as $[A]_\theta=\bigcap \{C: A\subset C\text{  and  }C\text{  is $\theta-$closed} \}$.
Now, my question is the following

Let $X$ be a topological space and $Y$ be a subspace of $X$, if $A\subset X$ and $G$ is the $\theta-$closed hull of $A\cap Y$ in the space $Y$. Is it true that $[A]_\theta \cap Y\subset G$?

Here, $[A]_\theta $ is the $\theta-$closed hull of $A$ in the space $X$.
I tried a lot to prove this or find a counter-example, and honestly, I feel like it is true, but I cannot justify it. Any comment is appreciated and thanks in advance.

REPLY [1 votes]: It is not true. It may happen that $x ∉ \operatorname{Cl}_θ^Y(A)$ but $x ∈ \operatorname{Cl}_θ^X(A)$, i.e. the point and the set are separated by disjoint neigborhoods from the point of view of the smaller space, but not from the point of view of the larger space.
The mininal (though not Hausdorff) counter-example is as follows: $X = \{a, b, c\}$, $a$ and $b$ are isolated, but only neighborhood of $c$ is the whole $X$, schematically: ((a) c (b)). And we put $Y := \{a, b\}$.
Another idea: Let $X$ be any $T_1$ space (but preferrably $T_2$) that is not regular. Then there is a point $x ∈ X$ and a closed set $A ⊆ X$ not containing $x$ such that $x ∈ \operatorname{Cl}_θ(A)$. But $A$ is $θ$-closed in $Y := \{x\} ∪ A$.
Also, in general every $θ$-closed set is closed. The other implication holds exactly in regular spaces. So your claim holds in regular spaces in a trivial way. On the other hand the claim does not hold in any non-regular $T_1$ space.
The Sierpiński two-point space is $T_0$ non-$T_1$ (and so non-regular), but your claim holds there just because it is two small. This can be in fact generalized – a space is hyperconnected (or anti-Hausdorff) if there are no disjoint nonempty open sets. These are exactly spaces with no non-trivial $θ$-closed sets. So your claim holds in spaces that are hereditarily hyperconnected – you won't find any non-trivial $θ$-closed subset in any subspace. These spaces are exactly hereditarily connected spaces, and equivalently nested spaces – those with linearly ordered topology (the topology itself is linearly ordered, do not confuse with linearly ordered topological spaces).
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