Suppose $P$ and $Q \subseteq P$ are posets, and let $\tau$ and $\rho$ be their respective Scott topologies. Now $Q$ is also equipped with the subspace topology $\tau\vert_Q$ inherited from $P$. It is easy to see that:

$$\tau\vert_Q \subseteq \rho$$

I have not found an example when the reverse inclusion is not also true. So my question is:

When do the two topologies on $Q$ coincide?


I have found an example of posets $P$ and $Q$, as above where:

$$\tau\vert_Q \neq\rho$$

Let $P=\mathcal{C}(\mathbb{N})$, the power set of the natural numbers ordered by set inclusion. Let $Q$ be the subposet consisting of cofinite elements of $P$. That is:

$$Q=\{x\in P : \complement x \ \mbox{is finite} \}$$

where $\complement x$ denotes the complement of $x$ in $\mathbb{N}$.

Now $Q$ clearly satisfies the Ascending Chain Condition. It follows that every directed set $S$ in $Q$ contains its supremum $\bigcup S$. To see this pick $x_0\in S$. If $x_0 = \bigcup S$, we are done. If not, pick $y\in S\setminus\downarrow\{x_0\}$. Since $S$ is directed there is $x_1\in S$ such that $x_0\subsetneq x_1$ and $y\subseteq x_1$. Repeating the process yields an ascending chain $x_0, x_1\ldots$ of elements that must be finite by ACC. So, for some $n$, $\bigcup S = x_n \in S$, as claimed.

Since every directed set in $Q$ contains its supremum, it follows that every up-set is open in its Scott topology $\rho$. So setting

$$B=\{\complement \{0\}, \mathbb{N}\}$$

we see that $B\in\rho$. I.e. it is open in the Scott topology on $Q$. Now $P$ is algebraic and its compact elements are precisely the finite subsets of $\mathbb{N}$. Thus:

$$\mathcal{B} = \{\uparrow\{x\} : x\in P \ \mbox{is finite}\}$$

is a basis for $\tau$, the Scott topology on $P$. Clearly each element of the basis contains an element of the form $\complement \{n\}\in Q$ where $n\ge 1\in \mathbb{N}$, and the same must be true for any $C\in\tau$. Since $B$ does not contain such an element it follows that for all $C\in\tau$, $C\cap Q\neq B$. We conclude that the two topologies $\rho$ and $\tau\vert_Q$ do not coincide.


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