Talk:Alexandrov topology

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The upward interior characterization given in the article is apparently wrong, because, if y is set to x, then one may conclude that any element in S is an interior point of S.

k, stuff about monotone added, will add brief mention of category theoretic gumpf about equivalence and bico-refection with "laymen" explanations. - 11 Oct, 2004

Will add the stuff about monotone = continous later ... 11 Oct 2004

Keep spelling with "-off" in references as these papers were published as such. - 9 Oct 2004

Ok, page moved back.

Some needed info

Who discovered that these spaces are the finitely generated objects in Top and what was the paper.
Who first used the Alexandrov topology (as opposed to the Scott topology or upward interval topology) in computer science. Scott? Plotkin? and where and when?
Who first used them in physics and where and when? Penrose?

- 8 Oct 2004

Please rename this Alexandroff topology the spelling with -off is more standard. 7-Oct-2004

I don't think so - not in modern books anyway. Charles Matthews 05:42, 7 Oct 2004 (UTC)

A Topological space has an Alexandrov topology if and only if all intersections of open sets is open (not just finite ones).

If in a preorder we declare open any final section (upper set) an Alexandrov topology obtains, but any such "fine" topology can be viewed that way, just taking the specialization (pre)order.

Between Alexandrov spaces, a function is continous iff it is monotone.

So, in fact, there are NOT finite topologies, just its specialization (pre)orders. Which in turns means (by Henkin's embedding theorem) that Preorder is the first "order" (in the logic sense) language of topology (But this means: topology is not first (logical) order!)

The topology can be finite and the space not but if the space is Kolmogorov and has finite topology is, obviously, finite)

from cleanup

someone requested this be re-worked as to be understandable to laymen. rhyax 20:46, 4 Sep 2004 (UTC)

Copy edit/house style

I've removed many duplicated links; the house style is to make a wikilink just once (I've not enforced this everywhere). I've also worked on the format to display some of the functions anf functors.

If the functors T and W are adjoint functors, there is no reason not to add comments to that effect.

Charles Matthews 07:57, 13 Oct 2004 (UTC)

Hmm well they are concrete isomorphisms which is stronger than being adjoint, there is in fact more going as we have bi-coreflection which I will say something about all in good time (I'm meant to be working not editing wiki pages :) - 13 Oct 2004

Name confusion: there are 2 topologies associated with GRT, both named Alexandrov

Please note that this Alexandrov-Topology and the topology for GRT originally introduced by Hawkings et al in "A new topology for curved space–time which incorporates the causal, differential, and conformal structures J. Math. Phys. 17, 174 (1976); DOI:10.1063/1.522874" as Alexandov-Topology for causal sets have nothing in common besides their name. Hence IMHO the last paragraph that reads "Inspired by the use of Alexandrov topologies in computer science, applied mathematicians and physicists in the late 1990s began investigating the Alexandrov topology corresponding to causal sets which arise from a preorder defined on spacetime modeling causality." is misleading, despite that some Authors -without noting the confusion of names nor mentioning the latter- indeed have used the first one in the context of space-time modeling. To clarify the paragraph should state that there on may associate two different topologies with GRT, both named "Alexandrov", one defined by future sets -in this case an Alexandrov-Topology as described in this article-, the second defined by the intersection of timelike cones.

A definition using intersections of past and future cones for GRT was given even earlier than (Hawking et al, 1976). It is discussed, for example, in E. H. Kronheimer & R. Penrose "On the structure of causal spaces", Proc. Camb. Phil. Soc. (1967), vol. 63, pp. 481-501.

131.111.145.118 (talk) 11:48, 22 June 2012 (UTC)[reply]

In an Alexandrov space the finite restriction is not relaxed.

In topology, an Alexandrov space (or Alexandrov-discrete space) is a topological space in which the intersection of any family of open sets is open. It is an axiom of topology that the intersection of any finite family of open sets is open. In an Alexandrov space the finite restriction is relaxed.

I think it is not accurate to say "relaxed". Any Alexandrov space is topology space, but not any topology space is Alexandrov space. The restriction is obviously tightened, not relaxed. Actually just the precondition is relaxed, but the property itself is tightened. —Preceding unsigned comment added by Fantadox (talk • contribs) 15:17, 25 November 2009 (UTC)[reply]

I disagree with writing "the finite restriction is strengthened". The property is strengthened, but "the finite restriction" is the hypothesis of the property, which is being relaxed. —Preceding unsigned comment added by 203.97.79.114 (talk) 02:08, 7 August 2010 (UTC)[reply]

Why Duality?

Why does the page say there's a *duality* between Alexandrov topological spaces and preorders when there's an equivalence, even isomorphism of categories?

Usually a duality would mean that there's an equivalence between one category and the opposite of the other. — Preceding unsigned comment added by 129.10.110.48 (talk) 14:36, 30 June 2017 (UTC)[reply]

Upward interior characterization

As already noticed in the talk page in 2004, the characterization by "upward interior" as written is nonsense. It should really say something like: A point x lies in the interior of a subset S of X iff S contains $\uparrow x$ , the principal upper set with respect to the specialization order. But even as modified, such a characterization is not saying anything very useful (it's basically a consequence of the "open up-set" characterization.) If nobody objects after a few days, I'll delete the entry for "upward interior". PatrickR2 (talk) 06:25, 15 July 2021 (UTC)[reply]

The upward interior characterization has been removed. PatrickR2 (talk) 04:02, 24 July 2021 (UTC)[reply]