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One can similarly see that the statement "Suprema of sequences are also topological limits, i.e., $\sup x_n=x \iff \lim x_n=x$" in the paper linked in your post is incorrect in general (and almost always). E.g., if $X=\mathbb R$ with the usual order and $x_n=0$ for all $n$, then $\sup x_n=0$, whereas the set of limits of $(x_n)$ in the Scott topology is $(-\infty,0]$.

One can similarly see that the statement "Suprema of sequences are also topological limits, i.e., $\sup x_n=x \iff \lim x_n=x$" in the paper linked in your post is incorrect in general (and almost always). E.g., if $X=(-\infty,0]$ with the usual order and $x_n=0$ for all $n$, then $\sup x_n=0$, whereas the set of limits of $(x_n)$ in the Scott topology is the entire set $X=(-\infty,0]$.

Details:

One can similarly see that the statement "Suprema of sequences are also topological limits, i.e., $\sup x_n=x \iff \lim x_n=x$" in the paper linked in your post is incorrect in general (and almost always). E.g., if $X=\mathbb R$ with the usual order and $x_n=0$ for all $n$, then $\sup x_n=0$, whereas the set of limits of $(x_n)$ in the Scott topology is $(-\infty,0]$.

Details:

• A point $b\in X$ is an upper bound of a subset $A$ of $X$ if $x\le b$ for all $x\in A$.

• A nonempty subset $D$ of $X$ is directed if any finite subset of $D$ has an upper bound in $D$.

• A point $s\in X$ is the supremum of a subset $A$ of $X$ and denoted by $\sup A$ or $\bigvee A$ if $s$ is the least upper bound of $A$, that is, if $s$ is an upper bound of $A$ and for any upper bound $b$ of $A$ we have $s\le b$.

• A subset $A$ of $X$ is an upper set if for any $x\in A$ we have $U_x\subseteq A$, where $U_x:=\{y\in X\colon x\le y\}$.

• A subset $A$ of $X$ is a lower set if for any $x\in A$ we have $L_x\subseteq A$, where $L_x:=\{y\in X\colon y\le x\}$.

• The pair $(X,\le)$ of a set $X$ with a partial order $\le$ is a directed-complete partial order (dcpo) if each directed subset $D$ of $X$ has a supremum. In what follows, $(X,\le)$ will be a dcpo.

• A subset $G$ of $X$ is open in the Scott topology, or Scott-open, if $G$ is an upper set and for any directed subset $D$ of $X$ such that $\sup D\in G$ we have $D\cap G\ne\emptyset$. Equivalently, a subset $F$ of $X$ is closed in the Scott topology, or Scott-closed, if $F$ is a lower set and for any directed subset $D$ of $F$ we have $\sup D\in F$.

-- The identity net $I_D$ on a directed subset $D$ of $X$ is the map $D\ni x\mapsto I_D(x):=x\in X$.

-- The set $\LL_D$ of limits of the identity net $I_D$ is defined by the following condition on $x\in X$:

• A point $b\in X$ is an upper bound of a subset $A$ of $X$ if $x\le b$ for all $x\in A$.

• A nonempty subset $D$ of $X$ is directed if any finite subset of $D$ has an upper bound in $D$.

• A point $s\in X$ is the supremum of a subset $A$ of $X$ and denoted by $\sup A$ or $\bigvee A$ if $s$ is the least upper bound of $A$, that is, if $s$ is an upper bound of $A$ and for any upper bound $b$ of $A$ we have $s\le b$.

• A subset $A$ of $X$ is an upper set if for any $x\in A$ we have $U_x\subseteq A$, where $U_x:=\{y\in X\colon x\le y\}$.

• A subset $A$ of $X$ is a lower set if for any $x\in A$ we have $L_x\subseteq A$, where $L_x:=\{y\in X\colon y\le x\}$.

• The pair $(X,\le)$ of a set $X$ with a partial order $\le$ is a directed-complete partial order (dcpo) if each directed subset $D$ of $X$ has a supremum. In what follows, $(X,\le)$ will be a dcpo.

• A subset $G$ of $X$ is open in the Scott topology, or Scott-open, if $G$ is an upper set and for any directed subset $D$ of $X$ such that $\sup D\in G$ we have $D\cap G\ne\emptyset$. Equivalently, a subset $F$ of $X$ is closed in the Scott topology, or Scott-closed, if $F$ is a lower set and for any directed subset $D$ of $F$ we have $\sup D\in F$.

• The identity net $I_D$ on a directed subset $D$ of $X$ is the map $D\ni x\mapsto I_D(x):=x\in X$.

• The set $\LL_D$ of limits of the identity net $I_D$ is defined by the following condition on $x\in X$: