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.
Best Answer
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 $(0,0)$.