Topological understanding of real Analysis I am almost done with my introductory class in Real Analysis for my undergrad and I am having a bit of difficulty grasping the idea of Basic Topology in R and applying it in Real Analysis. I studied alongside my course using 'Understanding Analysis 2ed' by Abott, but I am still lost... For instance I get what Open, Closed and Compact sets are, but when it comes to using it for proofs and in relation to continuity, I have no idea how to use the concepts at all. How did you guys approach it? Are there maybe online lectures that are good?
Thanks!
 A: The way I see it you have two options :

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*Forget about the word topology for the time being. Understand open and closed sets in $\mathbb{R}$ (or $\mathbb{R}^d$) via the definition of open balls, that is a set $A\subset\mathbb{R}^d$ is open if for all $x\in \mathbb{R}^d$ there exists $r>0$ such that $B_r(x) \subset A$, where $B_r(x)$ is the open ball of radius $r$ around $x$
$$ 
B_r(x):=\{y\in\mathbb{R}^d~:~|x-y|<r\}.
$$
Take the definition of a continuous function as the "$\epsilon,\delta$" definition which you will of learnt in your Analysis course. Continue along using these definitions / concepts, until you realise you need topology, then take a course in it.

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*If you have some spare time learn topology now. If you want to just get a quick idea of the point of it I recommend this fantastic $<1.5$ hour lecture : https://www.youtube.com/watch?v=7G4SqIboeig&list=PLFeEvEPtX_0S6vxxiiNPrJbLu9aK1UVC_&index=1&ab_channel=TheWE-HeraeusInternationalWinterSchoolonGravityandLight . In short : given a set $M$ we can define a topology $\mathcal{O}$ on $M$, which must satisfy three conditions (look them up). Once this CHOICE of $\mathcal{O}$ is made we call $(M,\mathcal{O})$ a topological space. The elements of $\mathcal{O}$ are subsets of $M$ and are called the open sets of $M$.



So the notion of open sets on $\mathbb{R}$ that you first learn in Analysis is ONE way of viewing open sets, this coincides with equipping $\mathbb{R}$ with whats called the Standard Topology, sometimes written $\mathcal{O}_{standerd}$.
Note people frequently work with the standard topology and hence don't explicitly say that they are working with it.
The whole point of topology is to add additional structure to your space which allows you to talk about continuity. Since the continuity of a function is directly related to the notions of open and closed sets we need to state which topologies we are working with. A function may be continuous with respect to one topology but not continuous with respect to another topology! Look up the topological definition of continuity.
