Applications of $\sf ZFC$/Set Theory in algebraic topology I'm doing this course in applications of set theory where we are learning about $\sf ZFC$, ordinals, cardinals, continuum hypothesis, Martin axioms, forcing, etc. At the end of the semester I must present a lecture in a topic that I like that has to do with this stuff.
I was wondering, is there any cool applications of axiomatic set theory to algebraic topology? I'll start studying the subject in the near future and I was wondering if I could merge those into something.
 A: I think the coolest application of axiomatic set theory in algebraic topology is the theorem of Casacuberta, Scevenels and Smith (http://www.ub.edu/topologia/casacuberta/articles/css.pdf).  It had been a fairly long-standing open question whether a localisation functor exists for every cohomology theory, and they showed that if you assume the large cardinal axiom Vopenka's Principle, the answer is yes.  Essentially, it was known how to localise with respect to a set, but localising with respect to a proper class raises foundational questions.  The (very vague!) gist of the proof is that assuming Vopenka's Principle, every class can be generated in a suitable sense by just a set, and then the known localisation machinery can be put to work.  Subsequent work by Bagaria, Casacuberta, Mathias and Rosicky (http://www.ub.edu/topologia/casacuberta/articles/bcmr.pdf) showed that a weaker large cardinal axiom suffices - the existence of a proper class of supercompact cardinals.
Having said all that, the details involve some fairly high-powered algebraic topology, so if you're just starting out it might be a struggle getting much beyond this vague overview for your project.  If that's the case, Hanul Jeon's suggestion might be a good one (if I do say so myself!).  When you start studying algebraic topology, the category of CW-complexes is often presented as a good category of "nice spaces" to work in.  One wrinkle, however, is that when you take the product of two CW complexes, the usual product topology is not necessarily the right topology to make it a CW complex, although in most situations you care about the two topologies are the same.  You might naturally ask when exactly the two topologies are the same; I found (https://arxiv.org/abs/1710.05296) that you can give a precise characterisation of when they're the same, and it depends on the bounding number, a naturally defined uncountable cardinal which can consistently be strictly less than the cardinality of the reals.  The result is certainly more point-set topology than modern mainstream algebraic topology, but the statement, relevant definitions, and with some effort the proof should be accessible to you.
