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I was wondering if the following is true: In a topological space with partial order, the inf and sup for a closed subset are achievable inside the subset so that they become minimum and maximum.

Is it still true if I replace "a topological space with partial order" with "a topological space with total order"?

In a topological space, what other kinds of condition can make inf and sup of a subset achievable in itself?

Thanks and regards!


More question:

In an "ordered" (not sure what kinds of order is proper here) topological space, are inf and sup of a subset accumulation points of the subset?


More questions again:

In Euclidean space, are inf and sup for a closed subset inside the subset? Are they accumulation points of the subset? What if in metric space? Thanks!

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    If you have a metric on a linearly ordered topological space, the required property is called "completeness".2010-08-17

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1 and 2: No. Take an interval $(\alpha, \beta)$ in $\mathbb{Q}$ where $\alpha, \beta$ are irrational. You can verify that such an interval is closed and that it contains neither its infimum nor its supremum. The property you want, for totally ordered sets, is called completeness.

3: I will assume you mean a totally ordered set with its order topology. In that case, this is not always true if the infimum or supremum are contained in the subset but true otherwise; this is just a matter of working through the definitions.

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    Thanks! For 3, why sup and inf are not accumulation points when they are in the subset?2010-08-17
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    Just take points that are isolated; e.g., in R, take {0,1}. Sup is 1, but is not an accumulation point; the inf is 0, but is not an accumulation point.2010-08-17
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    Sorry, instead of "false" I meant to say "not always true." Consider for example the infimum and supremum of a closed interval [a, b] in Z.2010-08-17
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    Can you please point out what form of order completeness ensures sup and inf are in the subset? The wikipedia page seems overwhelming to me. Thanks!2010-08-17
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    In Euclidean space, are inf and sup for a closed subset inside the subset? Are they accumulation points of the subset? What if in metric space? Thanks!2010-08-17
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    You might want to think about the examples you've been given already. Both the example I give above and then one given by Qiaochu clearly tell you that in Euclidean space the inf and sup may or may not be accumulation points. And R is also a metric space, so why do you think the examples don't answer your further questions?2010-08-17
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    Sorry... I just realized that. How about this question: In Euclidean space, are inf and sup for a closed subset inside the subset?2010-08-17
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    @Tim: what do you mean by infimum and supremum on R^n? With respect to what order?2010-08-17
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    Sorry, just in real field.2010-08-17
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    @Tim: in R, the infimum and supremum of a closed subset, when they exist, are in that subset. This is a consequence of the fact that R is complete (in both the order-theoretic and metric senses).2010-08-18
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    Thanks! Can you please point out what form of order completeness ensures sup and inf are in the subset? The wikipedia page you linked seems overwhelming to me.2010-08-18
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    @Tim: I have changed the link in the answer to the notion of completeness that is relevant to totally ordered sets. Note that you need some form of completeness to ensure that infima and suprema even exist.2010-08-18