## Basic questions:limits,reals and open sets [closed]

I am trying to gain a deeper understanding of limits and why they exist. Tumbling down the rabbit hole,i ended up face to face with the wikipedia definiton of open set.

Intuitively, an open set provides a method to distinguish between two points(using some metric) I don't understand how this works for Reals.Reals are infinitely divisible.Wikipedia uses The normal Euclidean metric to define a metric on Reals.While,it makes common sense to say that the distance between 2.5 and 3 is 0.5 and the distance between 2.5 and 4 is 1.5,the infinitely divisible nature of reals seem to suggest that there are infinite number of points between any 2 defined positions,which means 2.5,3 and 4 are all at infinity from each other.

Where am i going wrong?

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Posting on MO. This isn't really the place for what you're asking. I will give you some suggestions on books to take a look at. Take a look at "General Topology by Kelley", "Topology" by Munkres, "Principles of Mathematical Analysis" by Rudin. – Harry Gindi Mar 4 2010 at 5:07
You've just hit on a paradox: a seeming contradiction, when actually there is nothing wrong. Those two notions of infinity are completely different. You may also be interested in Zeno's paradoxes, if you haven't seen them: en.wikipedia.org/wiki/Zeno%27s_paradoxes. I'm not sure what open sets have to do with anything. If you really want a better understanding of open sets and of properties of the real numbers, learning some basic real analysis and general topology might be the way to go, as fpqc suggests. At this site questions that interest research mathematicians are preferred. – Jonas Meyer Mar 4 2010 at 5:17
@fpqc:Thank you for the recommendations.Will take a look at those books.I was planning to read up on point-set topology for quite some time now.I think i will get down to it.I was just wondering whether i made overlooked something trivial in my above reasoning.Thanks anyways. – Thiagarajan Mar 4 2010 at 5:17
A website where you might ask questions at this level (or find answers in the archive) is "Ask a topologist": at.yorku.ca/cgi-bin/bbqa – Peter Arndt Mar 4 2010 at 18:37
@Thiagarajan. Spend some time with the first three chapters of G. F. Simmons' book, "Introduction to Topology and Modern Analysis". – Regenbogen Mar 4 2010 at 18:48

## closed as off topic by David Speyer, Gjergji Zaimi, Joel David Hamkins, Reid Barton, Tom LeinsterMar 4 2010 at 23:51

I do agree with fpqc that this perhaps isn't the place to ask this question. However, I will answer it anyways, and give some references.

First of all, open sets don't really provide a way to distinguish between two points. For example, in a non-Hausdorff topology (see Munkres "Topology"), given two points $p \neq q$ in your space, one cannot always find open sets $U$ and $V$ containing $p$ and $q$ resp. such that $U \cap V$ is empty.

In the case of the real numbers $\mathbb{R}$, any open set in the metric topology really just gives us a way to take limits; that is, around every point in an open set $U$, there is a smaller open ball around the point contained in it. You'll find, I think, that many mathematicians take a lot of their topological intuition from this "analytic" topology.

A good introductory analysis text like Spivak's "Calculus" will help with your question.

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I disagree with your measure theoretic interpretation of the distance between 2 real numbers. Yes, the distance between a and b does happen to be the Lebesgue measure of [a,b], but measures are not needed (nor should they be used) to define a distance; that is what metrics are for. It is a coincidence. – Jonas Meyer Mar 4 2010 at 6:40
I wouldn't say that they're completely unrelated, since the open sets in the metric topology generates the Borel algebra, but I can see now that measure theory was perhaps not the best way to look at this. I was simply trying to explain that there are different mathematical ways to measure the size of sets (and hence distances) depending one how you want to look at it. I'm going to leave that part up in hopes that it is, at least, a little enlightening. – BMann Mar 4 2010 at 6:49
I agree that they are not completely unrelated, but it is a coincidence that only works for the special case of R that distance can be determined by measuring intervals. And there is no reason to do so, because you subtract the endpoints to find the distance. There is no general way to use measure theory to describe the distance between two points; that is what metrics do. – Jonas Meyer Mar 4 2010 at 6:58
The lebesgue measure is just the canonical Haar measure normalized to 1 on the unit interval, so it's even less measure-theoretic. – Harry Gindi Mar 4 2010 at 8:09
Fair enough. Post edited to remove the measure theory stuff. – BMann Mar 4 2010 at 18:35