Talk:Linear continuum

wikipedia is not a textbook

Some parts of this article are written like a textbook, not an encyclopedia article. The following bit, for example. --24.85.86.78 (talk) 05:15, 30 September 2011 (UTC)[reply]

Even though linear continua are important in the study of ordered sets, they do have applications in the mathematical field of topology. In fact, we will prove that an ordered set in the order topology is connected if and only if it is a linear continuum (notice the 'if and only if' part). We will prove one implication, and leave the other one as an exercise. (Munkres explains the second part of the proof[1])

I'd like to help improve this page. Any suggestions on how to better present this information? Siddharthist (talk) 06:44, 13 May 2017 (UTC)[reply]

Poorly written in places

A portion of the definition of the subject of this article is not well written. I have highlighted one word with boldface:

"Formally, a linear continuum is a linearly ordered set S of more than one element that is densely ordered, i.e., between any two distinct elements there is another (and hence infinitely many others), and which "lacks gaps" in the sense that every non-empty subset with an upper bound has a least upper bound. More symbolically:

"a) S has the least-upper-bound property

"b) For each x in S and each y in S with x < y, there exists z in S such that x < z < y

"A set has the least upper bound property, if every nonempty subset of the set that is bounded above has a least upper bound."

It is not clear from this explanation of the lesat upper bound property whether thehighlighted instance of the word "has" refers to "has" a least upper bound in itself — which is what it seems to say — or whether it refers to "has" in the larger set ".

(Of course to anyone who has already studied the real numbers, the intended meaning will be obvious; It can also possibly be guessed by reading further into the article. But neither of these facts justifies an unclear definition.)

Also, a minor point: If points a) and b) are supposed to begin with a capital letter (which we can see from b)), then they are sentences and should each end with period. Perhaps more graceful would be if point a) ended with a semicolon and point b) were left uncapitalized and ended with a period.Daqu (talk) 05:34, 4 October 2016 (UTC)[reply]