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In the book of General Topology by Munkress, at page 100, question 5, it is asked that

Let $X$ be an ordered set in the order topology. Show that $Cl(a,b) \subseteq [a,b]$. Under what condition the equality holds ?

I have proved that first part, and for why the inequality does not hold, I gave the following argument:

Consider $x=a$, then if there exist $c \in [a,b]$ such that $a<c$ and there is not other element between $a$ and $c$, then we have $(-\infty, c)\cap ([a,b]-\{a\}) = \emptyset$, hence by our previous statement in the theorem, $(-\infty, c)\cap ((a,b)'-\{a\}) = \emptyset$, and since $a\not \in (a,b)$, we have $a \not \in Cl(a,b).$

Therefore, the inequality hold iff $X$ is dense, hence for any $a,c \in X$, there are infinitely many elements between $a$ and $c$ in $X$.

So, is my argument correct ? or is there any problem with it ?

Edit: I have seen this answer after I have posted my question, but in there the answer giving as $a$ not having any left neighborhood, which is different from my answer.

Our
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  • If $a<b$ then $\overline {(a,b)}=[a,b]$ iff $(;a=\inf {x:x>a}$ and $b=\sup {y:y<b;).$ – DanielWainfleet Nov 22 '17 at 08:45
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    Dense is too strong of a condition. – William Elliot Nov 22 '17 at 08:51
  • @DanielWainfleet. In order theory terms iff a is not covered and b is not a cover. – William Elliot Nov 22 '17 at 08:53
  • @DanielWainfleet I do not understand what you trying to say, could you elaborate, please ? – Our Nov 22 '17 at 08:55
  • For any $u\in X$ let $(u,\to)$ denote ${v:v>u} $ and let $[u,\to)$ denote ${v:v\ge u}$ (regardless of whether or not $X$ has a largest member)......For $a<b$ the set $(a,\to)$ is not empty (because it contains $b$), and $a$ is a lower bound for $(a,\to).$.... Now if $a \ne \inf (a,\to)$ then there is a lower bound $c$ for $(a,\to)$ with $c>a.$ Then $c$ is also a lower bound for $(a,b)$ because $(a,b)\subset (a,\to). $ ..... So, since $a<c ,$ we have $a\not \in [c,\to) =$ $\overline { [c,\to)} \supset $ $\overline {(a,b)},$ so $a \not \in \overline {(a,b)}.$ – DanielWainfleet Nov 22 '17 at 10:35
  • @bof I is just a typo about the formatting, I corrected now. – Our Nov 23 '17 at 04:50
  • Is the question asking what conditions the given elements $a$ and $b$ must satisfy in order for $\operatorname{Cl}(a,b)=[a,b]$ to hold, or is it asking what conditions the ordered set $X$ must satisfy in order for $\operatorname{Cl}(a,b)=[a,b]$ to hold for all $a$ and $b?$ – bof Nov 23 '17 at 05:04
  • @bof I have just copied the question from the book directly, so there was no specification about that, but still we can assume both cases. I mean we can consider first for the element, and then for the set. – Our Nov 23 '17 at 05:07

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