Talk:Projective line

Topological circle

"Topologically, it is again a circle." Is the real projective line not also a circle in terms of differentiable manifolds? 145.97.196.54 17:42, 12 January 2006 (UTC)

Well, yes, that follows. Charles Matthews 17:58, 12 January 2006 (UTC)

Saying something is diffeomorphically a circle is certainly stronger than saying it is topologically so. linas 00:00, 13 January 2006 (UTC)

Really. So how many smooth structures does the circle as topological manifold carry, then? (The answer is one and only one.) Don't confuse this with the gap between continuous and smooth mappings, which is a different point. Charles Matthews 08:16, 13 January 2006 (UTC)

I'm not arguing with you. I was just trying to imagine what the anonymous poster was objecting to/asking about. (I certainly walk around with a mindset so that whenever I read the word "topological", I assume the author implied "not smooth" as a corollary, unless stated otherwise. (This is a habit from reading about ergodicity/chaos theory). Perhaps this is what anon was thinking too.) linas 01:35, 14 January 2006 (UTC)

AIUI the projective line is NOT a topological circle: a circle is a metric object (distinct from, say, an ellipse) embedded in a metric plane, and it has a identifiable sides - inside and outside. A projective line need have no metric but, more importantly, even in the projective plane it has only one side. I would call it a topological loop (1-manifold) but not a circle. Or do topologists differ from other geometers in the sense that they give to the word "circle"? -- Cheers, Steelpillow 07:09, 27 May 2008 (UTC)

The real projective line

A few days a go I made some changes which Mct mht reverted. Following a discussion here, I propose to reinstate the majority of my changes (with some minor correction), but would appreciate some third party consensus before risking an edit war.-- Cheers, Steelpillow 09:42, 2 June 2008 (UTC)

• The section on the real projective line states, "It is given by projecting points in R² onto the unit circle and then identifying diametrically opposite points." This should more correctly begin with something like; "A (metrically) finite visualisation is obtained by ...", although I am not sure if "visualisation" is the correct word here. -- Cheers, Steelpillow 09:42, 2 June 2008 (UTC)

• The same section also states, "One may also think of gluing the two "ends" of the real line onto a new point ∞ resulting in a circle." And in the introduction to the article, we find; "The projective line may also be thought of as the line K together with an idealised point at infinity" This last statement is true of the real projective line, but in general projective geometry has no metric and hence no concept of infinity. I would suggest that it be moved down to the section on the Real projective line, and the two statements be merged near the start of the section, something along the lines of; "It may be thought of as the real line K together with an idealised point at infinity, which connects to both ends of K creating a closed loop or topological circle." -- Cheers, Steelpillow 09:42, 2 June 2008 (UTC)

i was asked to respond in a edit summary. so i'll do it once. your comments are amateurish/crankish, both on my talk page and here. the edit you made was a minor one. it amounts to inserting the phrase "A (metrically) finite visualisation...", which, in this context, is garbage. Mct mht (talk) 23:14, 6 June 2008 (UTC)

Thanks for the reply Mct mht, and for your high opinion of my abilities (actually you are right about the amateur bit, though I have apparently studied synthetic geometry in greater depth than some professionals; you might like to compare my comments with the approach taken in the main projective geometry article). At your suggestion I discussed my edit here with the main contributor. I take the point about "metrically". But there is also an issue of NPOV to be considered, however distasteful this might be to anyone personally. For this reason I will reinstate the bulk of my edit, but will delete the "metrical" reference. As for the "visualisation" aspect, I have always been unsure if this was the right word (see above) but nobody has suggested anything better - I might try "example" until someone does. I would hope that in future if anybody is still unhappy with my edit they would correct the bit they are unhappy with, rather than revert the good along with the bad - especially if the alternative POV is unfamiliar to them. I am not a very experienced editor, but I have tried to follow Wikiquette; if I have failed to then I must apologise. But, as they say, there is only one way to gain experience. -- Cheers, Steelpillow 10:29, 7 June 2008 (UTC)

I have to agree with Mct mht. It has been years and this article is still in poor condition. What is all this nonsense about projecting points in the plane onto the unit circle and identifying antipodal points, or quotienting out by a "subgroup" (subgroup of what, exactly, being unclear). Why don't you just start with the unit circle and perform that identification (even though in the case of the 1-sphere, you happen to just get a 1-sphere back)? Better yet, just define it as the set of all lines through the origin in R² with the quotient topology. Michael Lee Baker (talk) 18:13, 21 September 2013 (UTC)

Scope

According to WP:What is an article?#Article scope the article should refer to everything that comes under its title. Therefore the lede and first section have been expanded in scope to include projective lines over not only fields, but also rings.Rgdboer (talk) 03:53, 24 February 2015 (UTC)

I hate to point out the obvious, but perhaps you are confusing the words scope and title? By your own admission, you have changed the scope of the article. —Quondum 05:51, 24 February 2015 (UTC)

I have reverted Rgdboer and rewritten the lead to be less technical and including context. About Rgdboer: If the projective line over a ring must be considered in the generalization, this would give it a undue weight consider the projective line over a field as a special case of a projective line over a ring. In fact most people, that learn projective lines or work with them, never consider projective lines over a ring. D.Lazard (talk) 11:14, 24 February 2015 (UTC)

For the record, the reversion started with this edit. As a result the article makes false statements such as only one point is added to make a projective line (not true in case of rings) and that the projective line is some how an extension of a line (geometry) (for a Galois field K there is no "line" in sight). The editor does not acknowledge the importance of equivalence relation in the definition of a projective line, so a link was restored today. Mention of "undue weight" demeans my contribution as "flat earth" advocacy. Note that I added three WP:reliable sources to this article that is deficient in them, while the editor added none. His prejudice is evident in his final sentence above. Discussion expected.Rgdboer (talk) 21:20, 24 February 2015 (UTC)

Rgdboer, the scope of this article is a projective line over a field, whatever the title might be. In this light, some of what you've said here does not apply. —Quondum 16:09, 25 February 2015 (UTC)