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.