Monotonicity of semi closure of sets in generalized topological spaces

Question

Hi I just want to ask if anybody here can show that if A is a subset of B then the semi closure of A is a subset of the semi closure of B. I know it is true for closure but I want to be sure if it holds for semi closure as well

Answer 1

Let us denote by $\underline{A}$ the semi-closure of set $A$. Then (as you mentioned in your comment) $\underline{A}$ is the smallest semi-closed set containing the set $A$.

We want to show that if $A\subset B$ then $\underline{A}\subset \underline {B}$.

Now, $$A\subset B\subset\underline{B}$$ and this means that $\underline{B}$ is a semi-closed set containing $A$. But $\underline{A}$ is the smallest semi-closed set containing the set $A$ and therefore, $$A\subset\underline{A}\subset \underline{B}.$$ Hope this help.

Note: In fact, you can also verify that $A\subset\underline{A}\subset\overline{A}$ where $\overline{A}$ is closure of $A$. This give a relationship between semi-closure and the closure.

Answer 2

Given the definitions in the comments we note that semi-closed sets are closed under arbitrary intersections and $X$ is always semi-closed. This allows us to define the semi-closure $\overline{A}$ of $A \subset X$ as

$$\overline{A} = \bigcap \{C \text{ semi-closed }, A \subseteq C\}$$

Now if $A\subseteq B$, then any semiclosed set $C$ that contains $B$ also contains $A$. So

$$\{C \text{ semi-closed }, B \subseteq C\} \subseteq \{C \text{ semi-closed }, A \subseteq C\}$$

So $\overline{A} \subseteq \overline{B}$ as the intersection of a possibly smaller family of subsets is larger.