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1. • CommentRowNumber1.
   • CommentAuthorjdw
   • CommentTimeApr 19th 2025
   • (edited Apr 19th 2025)
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   Author: jdw
   Format: MarkdownItexThe nlab article on the [[Scott topology]] claims that a function between two posets is continuous w.r.t. the Scott topologies iff it preserves directed joins. All the proofs of this fact that I've found in the literature seem to be non-constructive (usually using that $X \setminus x\!\downarrow$ is Scott open and that its complement is $x\!\downarrow$). One example is [here](https://planetmath.org/ScottContinuous). However the nlab article sounds to me as if it's written from a constructive point of view. Does anyone know if the fact is true constructively and if so, how to prove it?

   The nlab article on the Scott topology claims that a function between two posets is continuous w.r.t. the Scott topologies iff it preserves directed joins. All the proofs of this fact that I’ve found in the literature seem to be non-constructive (usually using that is Scott open and that its complement is ). One example is here. However the nlab article sounds to me as if it’s written from a constructive point of view. Does anyone know if the fact is true constructively and if so, how to prove it?

2. • CommentRowNumber2.
   • CommentAuthorDean
   • CommentTimeApr 20th 2025
   • (edited Apr 20th 2025)
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   Author: Dean
   Format: MarkdownItexMaybe one of these three is of some use: (1) in the case of the set-function $\forall : \text{FrameofOpens}(X) \rightarrow \mathbb{S}$ sending $\text{1}_{\text{X}}$ to $\text{1}$ and all else to $\text{0}$, preservation of directed joints is equivalent to the compactness of $\text{X}$. (2) Compactness is also equivalent to the existence of a convergent point of any ultrafilter. An ultrafilter $U$ converges to a point $x$, $U \rightarrow x$, if it extends its neighborhood filter $N_x$. (3) I think it's possible you could obtain a result for frames of locally compact locales by first concentrating on compact ones. It's a guess I'm making because the points of $\text{X}$ should correspond to morphisms of locales $\mathbb{S}^\text{X} \rightarrow \mathbb{S}$ for compact locales. Otherwise read on- they're just guesses.

   Maybe one of these three is of some use:

   (1) in the case of the set-function sending to and all else to , preservation of directed joints is equivalent to the compactness of .

   (2) Compactness is also equivalent to the existence of a convergent point of any ultrafilter. An ultrafilter converges to a point , , if it extends its neighborhood filter .

   (3) I think it’s possible you could obtain a result for frames of locally compact locales by first concentrating on compact ones. It’s a guess I’m making because the points of should correspond to morphisms of locales for compact locales.

   Otherwise read on- they’re just guesses.

3. • CommentRowNumber3.
   • CommentAuthorjdw
   • CommentTimeApr 21st 2025
   • PermaLink
   Author: jdw
   Format: MarkdownItexDean, thank you very much for your reply. Unfortunately, I don't really see the connection to my questions, could you elaborate a bit more? I don't see how compactness is related to my issue for example. Also the case distinction in (1) and the ultrafilters in (2) sound very non-constructive to me.

   Dean, thank you very much for your reply. Unfortunately, I don’t really see the connection to my questions, could you elaborate a bit more? I don’t see how compactness is related to my issue for example. Also the case distinction in (1) and the ultrafilters in (2) sound very non-constructive to me.

4. • CommentRowNumber4.
   • CommentAuthorʇɐ
   • CommentTimeApr 21st 2025
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   Author: ʇɐ
   Format: MarkdownItexI can’t find an explicit statement anywhere (e.g. [Tom de Jong’s dissertation](https://arxiv.org/abs/2301.12405) doesn’t treat the Scott topology, and [a follow-up article which does](https://doi.org/10.1017/S0960129523000282) doesn’t feature this statement), but Paul Taylor’s [[Practical Foundations of Mathematics]] (which De Jong refers to for a constructive treatment of domain theory) suggests that it isn’t true: on on p. 142 he proves that Scott continuity is equivalent to preimages of Scott _closed_ sets being Scott closed, and p. 143 features the remark > Such subsets are said to be Scott-open, and they form a topology. For any Scott-continuous function $ f \colon X \to Y $ between dcpos, the inverse image of any open subset $ V \subseteq Y $ is open. The converse holds so long as there are enough Scott-open sets to make the specialisation order (Example 3.1.2(i)) coincide with the given order. unfortunately without proof. (De Jong’s article states that the specialization preorder on a dcpo doesn’t coincide with the original order for all dcpos unless excluded middle holds, though they coincide constructively for continuous dcpos (lemma 36).)

   I can’t find an explicit statement anywhere (e.g. Tom de Jong’s dissertation doesn’t treat the Scott topology, and a follow-up article which does doesn’t feature this statement), but Paul Taylor’s Practical Foundations of Mathematics (which De Jong refers to for a constructive treatment of domain theory) suggests that it isn’t true: on on p. 142 he proves that Scott continuity is equivalent to preimages of Scott closed sets being Scott closed, and p. 143 features the remark

   Such subsets are said to be Scott-open, and they form a topology. For any Scott-continuous function between dcpos, the inverse image of any open subset is open. The converse holds so long as there are enough Scott-open sets to make the specialisation order (Example 3.1.2(i)) coincide with the given order.

   unfortunately without proof. (De Jong’s article states that the specialization preorder on a dcpo doesn’t coincide with the original order for all dcpos unless excluded middle holds, though they coincide constructively for continuous dcpos (lemma 36).)