I'm currently working through Jech's Set Theory. It's slow to read but I can understand pretty much everything I come across if I sit down and think about it long enough.
However, I don't feel very familiar with the material. In my experience, the best way to get familiar with a subject is to do problems. But I find the problems in this text quite easy. Most of them follow from definitions and the only somewhat trickier questions are those for which hints are given (and these hints typically give away the whole problem).
Thus I am looking for some problems, preferably but not necessarily with solutions, which are "tricky" in the sense that they require some insight and aren't just a simple application of some theorem, but are also "elementary" (to a set theorist) in that they don't require you to know something more advanced like several different kinds of forcing.
The topics I'm trying to improve in are:
- Cofinality
- Well-founded recursion
- Topology of $\mathbb{R}$ (things like closed sets, perfect sets) and the Baire space $\omega^{\omega}$
- Cardinal arithmetic problems involving $\sf{SCH}$, $\sf{GCH}$, and maybe problems to do with cardinals in the absence of $\sf{AC}$
- Filters/Ultrafilters
- Boolean algebras
- Club/Stationary sets
I realize this question might be a bit unclear, so here's an example of some easy problems and a slightly harder one:
(Easy) Show that the number of open sets of reals is $\mathfrak{c}$.
(Easy) Show that the number of continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$ is $\mathfrak{c}$.
(Slightly trickier) Show that the number of open sets of reals equals the number of continuous functions $f:\mathbb{R}\rightarrow\mathbb{R}$.
I am more fond of the third problem because it is not immediately obvious that one solution uses the cardinality of the reals (or, at least, it would not be obvious to someone who knows only that rationals are dense in the reals, how continuous functions are defined, and that $|\mathbb{R}|=2^{\aleph_{0}}$). Hopefully this illustrates the sort of problems I am looking for.
One such book is Problems and Theorems in Classical Set Theory (Problem Books in Mathematics)