Definition talk:Linearly Ordered Space

The usual technique for denoting a topology is by $\left({X, \tau}\right)$. I am wary about adding entities to a definition without good reason. In this case I would have thought that the definition of $\tau$ itself should encompass the ordered nature of the underlying set, and from the point of view of the topology, once you have accepted the fact of $\tau$ and its method of construction, the ordering as such is not relevant. Thoughts? --prime mover (talk) 18:04, 29 October 2012 (UTC)

I think the ordering should be included, since the word "ordered" is in the title. It isn't suggested anywhere that a linearly ordered space is (just) a special kind of topological space, nor do I think it should be. Comments? --abcxyz (talk) 18:41, 29 October 2012 (UTC)

As long as it's not proven that the ordering can be recovered from the topology (I don't know if it can, in general), I think it should be mentioned. Otherwise, the sheer saying "order topology" would mean something different (in standard usage, a topology induced by some order, regardless which order that may be).

The critical decision is which of these definitions we desire. Hopefully, they coincide (which happens when we can recover $\preceq$ from $\tau$). With that said, I take the more sophisticated viewpoint that it depends on whether $\preceq$ is used explicitly in the result under consideration (otherwise, I can see messy stuff on "this result doesn't depend on the choice of $\preceq$" cluttering pages). More investigation is needed; in the mean time, I'd vote for "order topology" to mean "a topology induced by a total order", and "order topology for $\preceq$" if $\preceq$ is to be stressed. --Lord_Farin (talk) 21:49, 29 October 2012 (UTC)

Given $\tau$, one cannot, in general, determine $\preceq$, or even $\left\{{\preceq, \succeq}\right\}$ (e.g. finite set or $\Q$).

"Orderable topology" would probably be a better term for a topology that can be induced by some total ordering, should it come up (similar to "metrizable topology").

I do think it is a good idea to change Definition:Order Topology to say "topology on $X$ induced by $\preceq$" or something similar, and to put the term "order topology" in the "Also known as" section. --abcxyz (talk) 23:15, 29 October 2012 (UTC)

On that last point I disagree unless you can find a solid source out there that so names it. --prime mover (talk) 06:15, 30 October 2012 (UTC)

I do agree with abcxyz that there should be a reference to $\preceq$ in the name, especially when it can't be recovered from the topology (which, as rightly pointed out, will generally not be the case - an interesting branch of research may be to determine conditions under which the topology induces a unique ordering). --Lord_Farin (talk) 13:17, 30 October 2012 (UTC)

Can we have some citations (book-based) for this definition and its many aliases? --prime mover (talk) 18:13, 13 February 2013 (UTC)