Is set theory important for topology?
Absolutely, absolutely. I honestly think that a failure for kids to grasp the nuances of set theory is probably the one thing that inhibits them most. Notions like Zorn's lemma, axiom of choice, ordinal numbers, etc. are standard mathematical parlance yet are often breezed over by kids as "technical details".
I can not tell you how many times people have come to me with difficult problems in algebra or topology and I was able to answer them instantly. Why? I didn't have some topological/algebraic trick up my sleeve--I just noticed that the two spaces person X was trying to prove weren't isomorphic/homeomorphic were of different cardinalities! Never forget forgetful functors.
If I ever had the ability to institute one course any undergrad hoping to become a mathematician should take it would be "Category Theory AND SET THEORY for the Working Mathematician". Seriously, if you ever hope to get somewhere especially in point-set topology you better be damn comfortable with the prerequisite set theory.
EDIT: I would recommend reading the following series of notes by MSE frequenter (and fantastic expositor) Pete L. Clark, as these are as close to "Set Theory for the Working Mathematician" as you are going to get:
Finite, Countable, and Uncountable Sets
Order and Arithmetic of Cardinals
Arithmetic of Ordinals
Some Cardinality Questions
(Original page source)
There are concepts from set theory that are heavily used in Topology that go beyond what you describe as "the basics".
Functions, inverse images, and the like are, of course, very important.
Products and disjoint unions are used in many important constructions in topology. You need to know what an arbitrary product of sets is, for example.
The ideas from equivalence relations and quotient sets are important for discussions of quotient topologies, among other places.
Some constructions and properties will require you to know at least the rudiments of ordinals and cardinals, as well as cardinal arithmetic.
Some important examples will require you to understand about ordinal arithmetic.
Some arguments will require you to understand ordinal induction/transfinite induction.
There are important connections between non-trivial set theory and topology. For example, Tychonoff's Theorem (a purely topological statement) is equivalent to the Axiom of Choice.
So... don't skip the chapter. If nothing else, do you really think that the authors were just trying to pad the book by adding a useless/unnecessary chapter at the beginning?
There are some important examples in general topology that require a basic knowledge at least of well-orderings, the first two infinite cardinals, $\omega$ and $\omega_1$, and the cardinal $2^\omega=\mathfrak{c}$. A basic understanding of ordinals makes these ideas easier to talk about and use. You could certainly get started in topology without these things, but I’d strongly recommend not skipping them, though you might work on them concurrently with the first parts of the topology proper.
I could probably teach a respectable one-semester introductory course in point-set topology without using any of this beyond the distinction between countable and uncountable sets, but I’d prefer not to, and I definitely wouldn’t try to teach a one-year course without those concepts. In particular, though you didn’t mention it specifically, you really, really should understand Zorn’s lemma, preferably with at least one of its equivalent forms (Axiom of Choice, Well-Ordering Principle, Hausdorff Maximal Principle, etc.): at some point you’ll definitely need that, and not just for topology.