Question

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?

Answer 1

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.