Intuitive significance open sets (and software for learning topology?)

Question

I have just started to learn topology and I referred to some books and online lectures. The problem is that they all talk the same things and are missing the same things.

I want to know "what is the intuitive significance of open set that makes it so important to be studied?"

I read reasons like it helps to prove continuity of function, but can

some one put very very specific examples that only open sets can achieve those results and not closed sets.

If the open sets state that we can get infinitely close to limit but cannot achieve it.... than why we cannot achieve the same with close sets (Eg $X$), say $X - \epsilon$ for any value of $\epsilon$, kind of opposite of $\delta$ and $\epsilon$ in calculus.

I also wanted to know if there is some software (like sage,mathematica) to study topology, my search couldn't get me useful result.

Answer 1

Consider a given point $p_0$ in your space $X$. When $X$ happens to be ordinary $d$-dimensional space ${\mathbb R}^d$, and you want to say "for all points $p$ sufficiently near $p_0$" such and such is true, resp., should hold, then you can talk about a ball with center $p_0$ such for all points $p$ in this ball said claim is true; and maybe you are even able to specify the radius $\epsilon>0$ of this ball.

Now there are spaces $X$ where you don't have an a priori notion of distance at your disposal. In such a case the notion of a topology comes to your help. It encodes the relevant aspects of convergence in metric spaces without actually using a metric. Essential is the following: Some subsets $O\subset X$ get the tag "open". The small open sets containing the point $p_0$ are like small balls with center $p_0$. In this way, when you want to talk about "all points sufficiently near $p_0$" you say: There is an open set $U$ containing $p_0$ (also called an open neighborhood of $p_0$) where such and such is true.

An example: Consider the point $(0,0)\in{\mathbb R}^2$. The points $(x,0)$ with $|x|<\epsilon$ form a tiny segment $S$ with center $(0,0)$, but $S$ is not a neighborhood of $(0,0)$. The sequence $z_n:=\bigl(0,{1\over n}\bigr)$ $\ (n\geq 1)$ converges to $(0,0)$ even though not a single $z_n$ lies in $S$; and the function $(x,y)\mapsto {\rm sgn}\,y$ is discontinuous at $(0,0)$, even though it is constant ($\equiv0$) on $S$. The reason: A true neighborhood of $(0,0)$ would contain a full small disk with center $(0,0)$.

Answer 2

Since open sets and closed sets are dual notions none of those notions is better than the other. You can express all "open-sets-topology" using "closed-sets-topology" just by dualizing every "open" definition, theorem etc. And vice-versa.

So why the topology is axiomatized in terms of open sets? In my humble opinion there is no rational reason behind it. It just gained more popularity.

There were other attempts to the task of axiomatization of topology, one of them was in terms of the closure operation. See Kuratowski Closure Axioms on Wikipedia for more details.

Answer 3

I think the crux of most answers to your question would essentially be that open sets are fundamental to convey a feel of "closeness" or "nearness" of points in arbitrary sets. And as far as I know its this "nearness" concept on which elementary notions in analysis like "convergence" or "continuity" are based upon.And as Wikipedia's page on open sets will tell you, open sets serve as a measure for nearness of points in sets without having to employ any "distance" function explicitly. So the use of open sets to me would lie in the above facts rather than any explicit applications. Closed sets are mostly defined as "complements" of open sets. So the word "complement" must tell you something. Even in defining closed sets otherwise, one has to employ the notion of limit points which in turn uses the concept of a neighbourhood and that will again lead you to Mr.open set. And as for information about how topology and computer science cross-over, these links might help. They deal with what would be Computational Topology of which I have little experience. They are not per-Se used to "study" Topology, but they might be of some use to you,I hope.

http://people.rit.edu/wfbsma/topology_and_its_applications/links.html

http://en.wikipedia.org/wiki/Computational_topology

Answer 4

I think openness of sets is intuitively relevant since they have direct relations with terms like "supremum" and "infimum", which then relates directly to limits, giving you the difference between $\mathbb{Q}$ and $\mathbb{R}$.

So simply said, openness of sets relates to the difference between $\mathbb{Q}$ and $\mathbb{R}$, and more such differences.

However, the relevance is much bigger than that. What sets are open and closed is the entire thing you study in general topology. Using its insights, you can develop more advanced mathematics like algebraic topology which gives you relations between topological spaces and group theory.