Talk:Alexandrov topology
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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)
more readable?
[edit] from cleanup
someone requested this be re-worked as to be understandable to laymen. rhyax 20:46, 4 Sep 2004 (UTC)
[edit] 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
