Motivation for Topology study in Real Analysis I'm an engineering student trying to work out some Real Analysis to learn how to write proofs (Needed for my PhD thesis) and just to rekindle my Calculus fires. From what I see, Real Analysis is the study of the basics of Calculus and "constructing" Calculus ground up. 
Why is Topology needed in the endeavor? I can't grapple my Engineering brain around compact sets and finite subcovers and stuff. I just can't imagine what's happening without spending 2 hours and 4 coffees.
The only "use" of this I've seen till now is for proving All Cauchy sequences are convergent in certain spaces. 
Can I skip topology? If yes, what is the bare minimum I need to do?
 A: You should know the following theorems from calculus on $\Bbb{R}$:


*

*Extreme Value Theorem: A continuous function $f$ on a closed bounded interval attains its maximum and minimum value on that interval.

*A continuous function on a closed bounded interval is uniformly continuous

*Intermediate Value Theorem: Let $f$ be a continuous function on $\Bbb{R}$ such that there are real numbers $a,b$ for which $f(a) < 0$ and $f(b) > 0$. Then there exists $c \in (a,b)$ such that $f(c)= 0$.
The first two theorems make use of a topological property known as compactness. The Heine - Borel Theorem is what tells you that in $\Bbb{R}$ with Euclidean metric, being closed and bounded is equivalent to being compact. Number 3 makes use of the topological property that $\Bbb{R}$ with the euclidean metric is connected.
Number 2 is especially important to know why a continuous function on a closed bounded interval is Riemann Integrable; this should be enough motivation for you to study topology!

The following I think is a cool example of where things get gnarly in topology. In a metric space (in fact in a topological space that is at least $T_2$) this cannot happen but: 
Put the trivial topology on $\Bbb{R}$, the family $A_n = (0,\frac{1}{n})$ is a countable collection of non-empty compact sets such that the intersection of any finite sub-collection is non-empty, but clearly 
$$\bigcap_{n=1}^\infty A_n = \emptyset .$$
A: An excellent use of compactness is showing that continuous functions can be integrated on a bounded closed interval (or on a bounded closed box in higher dimensions).
This is some fundamental fact that is useful in Calculus.
And personally I find that many proofs in real analysis go much smoother, and are actually easier to understand, once you stop talking about $\epsilon$'s and $\delta$'s and speak about neighborhoods instead.
Of course, "easier to understand" requires that you have gotten used to the more abstract language.
Finally, there are situations where you don't have reasonable notions of distance anymore, but you still have topology.
I have no idea whether this actually comes up in engineering, though.
A: Being an engineer I know you must be very practical, and the other answers give some good practical uses for the material you mentioned, but I wanted to throw out another angle:

The learning experience itself --and not just the content-- is valuable!

Even practical thinking people would benefit from thinking of this type of mathematics as mental weightlifting. What mental muscle does mathematics strengthen? Well, at least logical thinking, creativity, and ability to learn unfamiliar concepts (among other things). We can safely say that all of these are valuable traits of engineers! :)
So, it makes me a little concerned to hear "I just can't" in this context. I'm absolutely sure you will develop your understanding for it (beware of underestimating the time it will take, though) and at the end of the day you'll be a slightly smarter and cooler person for having stuck in there and overcoming the mental challenge.
A: The part of topology that is useful in real analysis is called point set topology. The three relevant topics of that part of topology are 1) continuity, which you've already had in "regular" real analysis, 2) connectivity and 3) compactness.
Those are the topology topics likely to be of greatest use to an engineer.
