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I haven't dealt with the Open Half-Line Topology much and I'm really unsure of how to go about calculating the required sections below.

Let $A$ = (-$\infty$,4) $\cup$ [5,$\infty)$ be a subset of $(R,C)$ and $C$ is the Open Half-Line Topology.

(a.) $Cl(A)$.

Closed sets in this topology are of the form $(-\infty,a]$. Thus the smallest closed set that would contain $A$ is $R$.

(b.) $Ext(A)$.

The definition I am using for $Ext(A)$ is "the set of all points $x$ $\in$ $X$ for which there exists an open set $U$ such that $x$ $\in$ $U$ $\subseteq$ $X - A$.

We have also been told that $Ext(A) = Int(X - A)$.

Based on that I have $Ext(A) = (4,5)$. I am not confident that this is correct.

(c.) $Bd(A)$.

The definition I am using for $Bd(A)$ is "the set of all points $x$ $\in$ $X$ for which every open set containing x intersect $A$ and $X - A$.

I believe this is $Bd(A)$ = {$4,5$}.

(d.) $Int(A)$.

The definition I am using for $Int(A)$ is "the set of all points $x$ $\in$ $X$ for which there exists an open set $U$ such that all $x$ $\in$ $U$ $\subseteq$ $A$.

I am confused on this one since I am not confident that the above answers are correct.

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For the most part it seems correct. The last part then follows by taking the set that contains every open interval (a,b) such that (a,b) is not a proper subset of (4,5).Hope this helps.

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Since exterior, interior, and boundary are all pairwise disjoint, then $\mathbb{R}=Ext(A)\cup Bd(A)\cup Int(A)$. So, we know that any one point cannot be in more than one of these sets. Also, the $Cl(A)=Int(A)\cup Bd(A)$ and $Ext(A)=\mathbb{R}-Cl(A)$.

Let $A=(-\infty,4)\cup[5,\infty)$. $A$ is not open in $C$ since it is not in the form $(a,\infty)$. Likewise, $A$ is not closed in $C$ since it is not of the form $(-\infty,a]$.

(a) $Cl(A)=\mathbb{R}$ because $\mathbb{R}$ is the smallest closed set that contains $A$.

(b) $Ext(A)=\mathbb{R}-Cl(A)=\emptyset$.

(d) $Int(A)$ is the largest open set contained within $A$. If $A$ is open, then $Int(A)=A$. Since we established earlier that $A$ is not open, then the $Int(A)\subset A$. The largest open set of the form $(a,\infty)$ that is contained within $A$ is $(5,\infty)$. So, I believe $Int(A)=(5,\infty)$.

(c) For $Bd(A)$, the open sets you are using such that $x\in U$ and $U\cap A$ and $U\cap(X-A)$ should be considered open with respect to the topology $C$, so they should be of the form $(a,\infty)$. But, $Cl(A)=Int(A)\cup Bd(A)$, so $\mathbb{R}=(5,\infty)\cup Bd(A)$ and since $Bd(A)$ and $Int(A)$ are pairwise disjoint, so $Bd(A)=(-\infty,5]$.

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