<p>Since the least element $\bot$ of any Scott domain is a <em>compactification point</em> $-$ the only open set containing it is the whole space $-$ the Scott topology is never Hausdorff, unless it is trivial.</p>

<p>An information system $(A,Con,\vdash)$ induces a trivial domain iff it is itself trivial: $Con = \{\emptyset\}$.  Otherwise, there exists at least one point above $\bot$.</p>

<p>The total elements of the Scott domain can indeed be given their own topology sometimes, which may be $T_2$ or even metrizable.</p>

CC BY-SA 3.0
2017-04-12T18:35:35.013Z
There's a whole industry about the topology of the maximal elements of a domain.
Andrej Bauer
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2017-04-13T07:43:40.710Z
Thanks Andrew and @AndrejBauer . Okay, so the "take-away" seems to be that I should adjust my interest/thinking to total elements. Then the closest to my interests has been Edalat's https://www.doc.ic.ac.uk/~ae/papers/survey.ps , where the total/maximal elements of "formal ball domains" comprise a Hilbert (or other metric) space. But my broader background re "whole industry about the topology of maximal elements" is meager or non-existent. Can you suggest any introductory/survey material, particularly with a nice bibliography ( assuming background ~ http://dl.acm.org/citation.cfm?id=200952 )?
John Forkosh
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2017-04-13T08:33:08.043Z
I would recommend tracking down the work of [Keye Martin](http://dblp.uni-trier.de/pers/hd/m/Martin:Keye) and [Pawel  Waszkiewicz](http://dblp.uni-trier.de/pers/hd/w/Waszkiewicz:Pawel).
Andrej Bauer
CC BY-SA 3.0
2017-04-13T08:51:17.460Z
Not to take away from your takeaway -- which is probably correct -- I would also add that recently there has been significant progress in extending the classical theory of "nice" spaces to the non-Hausdorff setting.  
See [Quasi-Polish spaces](http://www.sciencedirect.com/science/article/pii/S0168007212001820) by Matthew de Brecht.
Andrew Polonsky
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2017-04-14T08:55:40.377Z
@AndrejBauer Thanks for the references. I've read Martin's "Partiality in Physics", and "A domain of spacetime intervals in general relativity" (and also http://arxiv.org/abs/gr-qc/0407093/ which seems to be missing from the uni-trier list), and browsed one or two of his papers on measurement in domains (including a few sections of his thesis). But I hadn't seen "The space of maximal elements in a compact domain", or maybe I came across it but failed to recognize its usefulness. Waszkiewicz is new to me, and I'll try to look over his bibliography more carefully. Thanks again.
John Forkosh
CC BY-SA 3.0
2017-04-14T09:47:04.957Z
@AndrewPolonsky Thanks for the reference. That paper indeed looks very interesting, both abstractly in its own right, and, like you suggest, maybe generalizing the spaces I should be looking at. The first five or so sections seem pretty much within my grasp, after which I hope to jump to section 11. Thanks again.
John Forkosh
CC BY-SA 3.0
2017-04-12T12:20:20.530Z
2017-04-12T14:42:31.580Z
2017-04-12T14:42:31.580Z
Andrew Polonsky
<p>As per subject, if a Scott domain with $T_2$ topology is to be constructed from a Scott information system, then what besides the usual definitional requirements, e.g., <a href="https://en.wikipedia.org/wiki/Scott_information_system" rel="noreferrer">https://en.wikipedia.org/wiki/Scott_information_system</a> , does the information system have to satisfy?</p>

<p>Information systems are more completely discussed in (Scott's original paper) <a href="https://www.researchgate.net/publication/220897586_Domains_for_Denotational_Semantics" rel="noreferrer">https://www.researchgate.net/publication/220897586_Domains_for_Denotational_Semantics</a> (note: the original paper is followed by a three-page bibliography missing from this pdf) containing a section "Topological Considerations" starting on page 586, where "Hausdorff" is briefly mentioned on page 588 with respect to (what Scott calls) "total elements", $Tot_A$, assuming no unique top. But the domain itself, what Scott calls $\mid A\mid$, is still $T_0$ in general. And I'm asking for additional definitional requirements so that $\mid A\mid$, when contsructed from the corresponding information system, is "automatically" $T_2$.</p>

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2017-04-12T18:50:17.013Z
As Andrew points out, the only way to get $T_2$ for the entire domain is to make it trivial. But also note that apart from Scott topology on a domain there is also the Lawson topology on a domain, which is Hausdorff.
Andrej Bauer
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2017-04-13T07:52:59.197Z
Thanks, @Kaveh for your comment about Stone duality, though you seem to have deleted it. Scott's "brief mention" that I cited above indeed also says  "compact totally disconnected Hausdorff space", and I'd imagine you're likely right that this suggests some connection with the Stone representation theorem (though google's not immediately coughing it up).
John Forkosh
CC BY-SA 3.0
2017-04-13T08:04:05.087Z
Thanks @AndrejBauer (and also thanks for comment below). And I think you may be right, as follows. If for every data token $a\in A$ there's a complement $a^\perp\in A$, maybe analogous to $a\subseteq\mathbb{N}\;\mbox{r.e.}\Longrightarrow \mathbb{N}\backslash a\;\mbox{r.e.}$, then would that induce a Lawson topology on the corresponding constructed domain??? (Haven't tried to answer that myself yet; only just now saw your comment and had the preceding thought.)
John Forkosh
CC BY-SA 3.0
2017-04-13T08:27:24.733Z
Note that the maximal set of an (algebraic) domain is totally disconnected, whereas Stone duality gives you zero-dimensional spaces. These are two different concepts, and indeed we get different kinds of spaces. There is no direct relation to Stone duality, but of course one can cook up a slightly less direct one (as always).
Andrej Bauer
CC BY-SA 3.0
2017-04-14T08:24:24.193Z
Thanks again, @AndrejBauer . I'm now thinking your and A.P.'s remarks below are likely setting me on the right track -- think about the space of total elements rather than the domain. That gives more opportunity/flexibility to (your words) "cook up" interesting stuff.
John Forkosh
<domain-theory>
What else (besides the usual) can be said about a Scott Information System if the constructed domain is required to be Hausdorff?
John Forkosh
# What else (besides the usual) can be said about a Scott Information System if the constructed domain is required to be Hausdorff?

(Asked by: username_0 on 2017-04-11)

As per subject, if a Scott domain with $T\_2$ topology is to be constructed from a Scott information system, then what besides the usual definitional requirements, e.g., <https://en.wikipedia.org/wiki/Scott_information_system> , does the information system have to satisfy?

Information systems are more completely discussed in (Scott's original paper) <https://www.researchgate.net/publication/220897586_Domains_for_Denotational_Semantics> (note: the original paper is followed by a three-page bibliography missing from this pdf) containing a section "Topological Considerations" starting on page 586, where "Hausdorff" is briefly mentioned on page 588 with respect to (what Scott calls) "total elements", $Tot\_A$, assuming no unique top. But the domain itself, what Scott calls $\mid A\mid$, is still $T\_0$ in general. And I'm asking for additional definitional requirements so that $\mid A\mid$, when contsructed from the corresponding information system, is "automatically" $T\_2$.

**Tags:** domain-theory
> **Comments:**
> **username_1**: As Andrew points out, the only way to get $T_2$ for the entire domain is to make it trivial. But also note that apart from Scott topology on a domain there is also the Lawson topology on a domain, which is Hausdorff.  
> **username_0**: Thanks, @Kaveh for your comment about Stone duality, though you seem to have deleted it. Scott's "brief mention" that I cited above indeed also says  "compact totally disconnected Hausdorff space", and I'd imagine you're likely right that this suggests some connection with the Stone representation theorem (though google's not immediately coughing it up).  
> **username_0**: Thanks @AndrejBauer (and also thanks for comment below). And I think you may be right, as follows. If for every data token $a\in A$ there's a complement $a^\perp\in A$, maybe analogous to $a\subseteq\mathbb{N}\;\mbox{r.e.}\Longrightarrow \mathbb{N}\backslash a\;\mbox{r.e.}$, then would that induce a Lawson topology on the corresponding constructed domain??? (Haven't tried to answer that myself yet; only just now saw your comment and had the preceding thought.)  
> **username_1**: Note that the maximal set of an (algebraic) domain is totally disconnected, whereas Stone duality gives you zero-dimensional spaces. These are two different concepts, and indeed we get different kinds of spaces. There is no direct relation to Stone duality, but of course one can cook up a slightly less direct one (as always).  
> **username_0**: Thanks again, @AndrejBauer . I'm now thinking your and A.P.'s remarks below are likely setting me on the right track -- think about the space of total elements rather than the domain. That gives more opportunity/flexibility to (your words) "cook up" interesting stuff.  

---

## 1 answers

### Answer by username_2 on 2017-04-12. Score: 10 | Accepted answer

Since the least element $\bot$ of any Scott domain is a *compactification point* $-$ the only open set containing it is the whole space $-$ the Scott topology is never Hausdorff, unless it is trivial.

An information system $(A,Con,\vdash)$ induces a trivial domain iff it is itself trivial: $Con = \{\emptyset\}$. Otherwise, there exists at least one point above $\bot$.

The total elements of the Scott domain can indeed be given their own topology sometimes, which may be $T\_2$ or even metrizable.

> **Comments:**
> **username_1**: There's a whole industry about the topology of the maximal elements of a domain.  
> **username_0**: Thanks Andrew and @AndrejBauer . Okay, so the "take-away" seems to be that I should adjust my interest/thinking to total elements. Then the closest to my interests has been Edalat's https://www.doc.ic.ac.uk/~ae/papers/survey.ps , where the total/maximal elements of "formal ball domains" comprise a Hilbert (or other metric) space. But my broader background re "whole industry about the topology of maximal elements" is meager or non-existent. Can you suggest any introductory/survey material, particularly with a nice bibliography ( assuming background ~ http://dl.acm.org/citation.cfm?id=200952 )?  
> **username_1**: I would recommend tracking down the work of [Keye Martin](http://dblp.uni-trier.de/pers/hd/m/Martin:Keye) and [Pawel  Waszkiewicz](http://dblp.uni-trier.de/pers/hd/w/Waszkiewicz:Pawel).  
> **username_2**: Not to take away from your takeaway -- which is probably correct -- I would also add that recently there has been significant progress in extending the classical theory of "nice" spaces to the non-Hausdorff setting.   See [Quasi-Polish spaces](http://www.sciencedirect.com/science/article/pii/S0168007212001820) by Matthew de Brecht.  
> **username_0**: @AndrejBauer Thanks for the references. I've read Martin's "Partiality in Physics", and "A domain of spacetime intervals in general relativity" (and also http://arxiv.org/abs/gr-qc/0407093/ which seems to be missing from the uni-trier list), and browsed one or two of his papers on measurement in domains (including a few sections of his thesis). But I hadn't seen "The space of maximal elements in a compact domain", or maybe I came across it but failed to recognize its usefulness. Waszkiewicz is new to me, and I'll try to look over his bibliography more carefully. Thanks again.  
> **username_0**: @AndrewPolonsky Thanks for the reference. That paper indeed looks very interesting, both abstractly in its own right, and, like you suggest, maybe generalizing the spaces I should be looking at. The first five or so sections seem pretty much within my grasp, after which I hope to jump to section 11. Thanks again.  

---

