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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

user402701
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    Define semi closure and generalized topological spaces, please – Henno Brandsma Feb 14 '17 at 11:46
  • The semi closure of a set is the smallest semi closed set containing the set. – user402701 Feb 14 '17 at 11:49
  • So, a set is semi-closed if its complement is semi-open? – Juniven Acapulco Feb 14 '17 at 11:51
  • A generalized topology is a collection of subsets of X such that the empty set is in the collection and that the union of the elements in the collection is also in the collection – user402701 Feb 14 '17 at 11:52
  • Yes. A set is semi closed if its complement is semi open – user402701 Feb 14 '17 at 11:53
  • I think it follows from the result that the union of semi open sets is semi open – user402701 Feb 14 '17 at 12:10
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    The given collection are then the semi-open sets? – Henno Brandsma Feb 14 '17 at 12:14
  • You have told us what a generalized topology ism but you haven't told us what the "semi-open sets" of a generalized topology are. – bof Feb 14 '17 at 12:23
  • Thanks. I have tried this one. What I am not very sure of is if the semi closure of A is a subset of B given that A is a subset of B – user402701 Feb 14 '17 at 12:25
  • It follows from *what* result that the union of semi open sets is semi open? I don't see any "results" here. – bof Feb 14 '17 at 12:26
  • A set is semi open in a generalized topological space if it is a subset of the closure of its interior – user402701 Feb 14 '17 at 12:27
  • @bof Maybe if you post that as a question, then that can be answered. I have a copy of that result. – Juniven Acapulco Feb 14 '17 at 12:42
  • How about giving the definitions in order, and putting them in the question, instead of in comments? How are "closure" and "interior" defined in a semitopological space>? How are "open sets" and "closed sets" defined? – bof Feb 14 '17 at 12:54
  • @ΘΣΦGenSan I'd just like to see some *definitions*. I understand that a "generalized topology" on $X$ is a collection of subsets that is clused under arbitrary union. Fine. What is the definition of an OPEN SET and a CLOSED SET is a generalized topological space? And then what ate the definitions of INTERIOR, CLOSURE, SEMI-OPEN SET, AND SEMI-CLOSED SET? – bof Feb 14 '17 at 13:20
  • @bof It's too long to put them in here. Sorry. But there many articles that cover those terms you mentioned. One good reference is N. Levine, “Semi-open sets and semi-continuity in topological spaces,” The American Mathematical Monthly, vol. 70, pp. 36–41, 1963. – Juniven Acapulco Feb 14 '17 at 13:28

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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.

Juniven Acapulco
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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.

Henno Brandsma
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