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I Believe the answer is when the set is closed in the sense of standard topology (we exclude $\mathbb{R}$) itself.

Examples:

  1. Point sets have a minimum, and they are closed.

  2. Closed interval also have a minimum, which is simply the lower bound $[a,b]->a$

If my guess is right, how do I prove it, else what is the correct characterization?

I'm guessing this has something to do with well orderedness. What I got so far from MSE:

  1. Well orderedness is preserved under subsets

  2. Well ordered subsets must be countable

Clemens Bartholdy
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    $(-\infty, 0]$ is closed, but it does not have a minimal element... – Julio Puerta Jun 16 '24 at 20:10
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    $[0,1)$ is not closed, but it does have a minimum element. – Lee Mosher Jun 16 '24 at 20:13
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    Suppose that $X\subseteq (0,\infty)$, then ${0}\cup X$ has a minimum and you can choose $X$ to be as complicated as you want it to be. – dialegou Jun 16 '24 at 20:17
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    also, being well ordered is not the same as having a minimum element. – Carlyle Jun 16 '24 at 20:17
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    There is no topological characterization: $[0, \infty)$ has a minimum element, $(-\infty), 0]$ doesn't, but $-$ is a homeomorphism of $\mathbb R$, which maps one to the other. Similar for well-ordered. – Ulli Jun 16 '24 at 20:25
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    I feel like this question got downvoted just because it does not have a very good answer. But that is not the fault of the OP. – Jonathan Hole Jun 16 '24 at 20:31
  • There isn't really a simpler answer than "having a minimum" itself. Being Euclidean closed and bounded below is certainly a sufficient condition, but by far not a necessary one. – Sassatelli Giulio Jun 16 '24 at 21:53
  • It's sufficient for $A \subseteq \mathbb{R}$ to be compact, but this is not necessary (the example $[0, 1)$ applies). – K. Jiang Jun 16 '24 at 21:58

1 Answers1

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This question unfortunately does not have an interesting answer. It also doesn’t have an answer based on the topology (as shown by @Ulli)

Here is the nicest description I could come up with : $$\{\{x\} \cup Y_x | x\in \mathbb{R}, Y_x \subset (x,\infty)\}$$ is the collection of subsets of $\mathbb{R}$ with a least element.

Malady
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