TITLE: What are necessary and sufficient conditions for a supremum to be a limit point?
QUESTION [4 upvotes]: A question in similar spirit has already been asked here, but was wrongly accepted. 
Consider the case $A = \{0, 1, 2\} \subset \mathbb{R}$, this has $\sup(A) = 2$, but $2$ is not a limit point of $A$.
Note that the answer to this question could be sensitive to the definition of limit, so I give the definition from (Croom 1989):
Let $(X,T)$ be a topological space and $A \subset X$, a point $x \in A$ is called a limit point of A if

every open set containing $x$ contains a point of $A$ distinct from A.

A sufficient condition would be that the supremum does not lays in $A$, e.g. $A = (0,1)$ has $\sup(A) = 1$ and $1$ is a limit point of $
A$.
But it is not a necesary condition (consider $[ 0, 1 ]$).
Can we say that the supremum is either a maximum or a limit point?

REPLY [1 votes]: Alternatively, let $X$ be a linearly ordered topological space and let $A ⊆ X$. There are five mutually exclussive possibilities regarding $\sup(A)$:

$A$ has no supremum.
$\sup(A) ∈ A$ and $\sup(A) ∈ \overline{A ∩ (←, \sup(A))}$: limit maximum.
$\sup(A) ∈ A$ and $\sup(A) ∉ \overline{A ∩ (←, \sup(A))}$: isolated maximum.
$\sup(A) ∉ A$ and $\sup(A) ∈ \overline{A ∩ (←, \sup(A))}$: non-attained supremum.
$\sup(A) ∉ A$ and $\sup(A) ∉ \overline{A ∩ (←, \sup(A))}$: $A = ∅$ and $\sup(A) = \min(X)$ … “empty supremum”.

All five cases are realized in $X = [0, ∞)$: $A$ takes the value $[0, ∞)$, $[0, 1]$, $\{0, 1\}$, $[0, 1)$, and $∅$, respectively.
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