I just posted this as a comment to one of the existing answers, but then thought it might deserve to be expanded and published as a separate answer.
There is an old set theory book by D.Monk,and a few days ago I realized he has on his website what he calls Lectures on set theory (also solutions for exercises) which seems to me like a new book, which, in terms of topics covered, is comparable to Jech and Kunen's texts (and may be preferable to some readers depending on taste). (Also, as seen below, it contains fairly recent results, e.g. a proof that $\frak p=\frak t$ (as well as PCF, which perhaps no longer counts as a recent topic, but is good to see in a book form).) I do not know if it has officially been (or will be) published, the pdf I see is marked March 11, 2019. See http://math.colorado.edu/~monkd/ and in particular http://euclid.colorado.edu/~monkd/setth.pdf
Update, I had overlooked a bigger and newer file on the same web page
http://euclid.colorado.edu/~monkd/full.pdf
NOTES ON SET THEORY J. Donald Monk September 14, 2019
Here are some excerpts (for the smaller, older file) :
Preface
Edition of March 11, 2019: chapter on $\frak p=\frak t$ rewritten.
Edition of August 9, 2017: chapter on proper forcing rewritten.
Edition of November 14, 2016: chapter on proper forcing changed; the proof of Theorem 28.5 was in error, and a new proof using a game is given (Theorem 28.33).
Edition of August 30, 2016: Proposition 27.21 corrected.
Some background on these notes:
0. The exercise solutions have not been carefully checked.
1. The axioms for first-order logic are due to Tarski.
2. The treatment of forcing follows Kunen, except for using Boolean values in the definition.
3. The proof of Hausdorff’s theorem in Chapter 17 follows Hausdorff’s original proof
closely.
4. The treatment of proper forcing in Chapter 28 follows Jech to a large extent.
5. For PCF in chapters 30–32 we follow Abraham and Magidor.
6. Chapter 33 is based on Blass.
7. The proof that $\frak p=\frak t$ in Chapter 34 is based upon notes of Fremlin and a thesis of
Roccasalvo.
8. The consistency proofs in Chapter 35 are partly from Kunen and partly from the author.
TABLE OF CONTENTS
LOGIC
1. Sentential logic .... 1
2. First-order logic .... 12
3. Proofs .... 24
4. The completeness theorem .... 42
ELEMENTARY SET THEORY
5. The axioms of set theory .... 67
6. Elementary set theory 70
7. Ordinals, I .... 75
8. Recursion .... 80
9. Ordinals, II .... 87
10. The axiom of choice. .... 106
11. The Banach-Tarski paradox .... 111
12. Cardinals .... 121
GENERIC SETS AND FORCING, I
13. Boolean algebras and forcing orders. .... 144
14. Models of set theory .... 160
15. Generic extensions and forcing . .... 186
16. Independence of CH .... 207
INFINITE COMBINATORICS
17. Linear orders .... 215
18. Trees .... 241
19. Clubs and stationary sets .... 253
20. Infinite combinatorics .... 267
21. Martin’s axiom .... 277
22. Large cardinals .... 288
23. Constructible sets. .... 311
GENERIC SETS AND FORCING, II
24. Powers of regular cardinals. .... 332
25. Isomorphims and $\neg$AC .... 340
26. Embeddings, iterated forcing, and Martin’s axiom .... 352
27. Various forcing orders. .... 369
28. Proper forcing .... 392
29. More examples of iterated forcing .... 412
PCF
30. Cofinality of posets .... 422
31. Basic properties of PCF .... 456
32. Main cofinality theorems. .... 478
CONTINUUM CARDINALS
33. $^\omega \omega$ and $\mathscr P(\omega)/fin$ .... 504
34. $\frak p=\frak t$ .... 521
35. Consistency results concerning $\mathscr P(\omega)/fin$ .... 556
APPENDICES
36. The integers .... 565
37. The rationals .... 572
38. The reals .... 578
INDEX OF SYMBOLS .... 588
INDEX OF WORDS .... 592
Update, contents for the bigger and newer file:
http://euclid.colorado.edu/~monkd/full.pdf
NOTES ON SET THEORY
J. Donald Monk September 14, 2019
A large portion of these notes, roughly Chapters 12–31, follows Kunen [2011].
TABLE OF CONTENTS
LOGIC
1. Sentential logic .... 1
2. First-order logic .... 16
3. Proofs .... 29
4. The completeness theorem .... 49
ELEMENTARY SET THEORY
5. The axioms of set theory .... 77
6. Elementary set theory .... 80
7. Ordinals, I .... 89
8. Recursion .... 94
9. Ordinals, II .... 103
10. The axiom of choice. .... 127
11. Cardinals .... 135
MODELS OF SET THEORY
12. The set-theoretical hierarchy .... 164
13. Absoluteness. .... 184
14. Checking the axioms .... 194
15. Reflection theorems. ... 200
16. Consistency of no inaccessibles. .... 207
17. Constructible sets. .... 208
INFINITE COMBINATORICS
18. Real numbers in set theory .... 225
19. The Cicho´n diagram .... 276
20. Continuum cardinals .... 295
21. Linear orders .... 320
22. Trees .... 358
23. Clubs and stationary sets .... 394
24. Infinite combinatorics ... 427
25. Martin’s axiom .... 438
26. Large cardinals .... 480
FORCING
27. Boolean algebras and forcing orders. .... 509
28. Generic extensions and forcing .... 542
29. Forcing and cardinal arithmetic .... 566
30. General theory of forcing .... 581
31. Iterated forcing .... 610
32. $\frak p=\frak t$ .... 694
PCF
33. Cofinality of posets .... 733
34. Basic properties of PCF .... 766
35. Main cofinality theorems .... 787
ADDITIONAL SECTIONS
36. Various forcing orders .... 813
37. More examples of iterated forcing .... 836
38. Consistency results concerning $\mathscr P(\omega)/fin$ .... 846
REAL NUMBERS
39. The integers .... 855
40. The rationals .... 862
41. The reals .... 868
Index of symbols .... 878
Index of words .... 885
Jan. 13, 2019: indices corrected
March 11, 2019: chapter on $\frak p=\frak t$ rewritten.
June 11, 2019: correction concerning $L[B]$.
Sept. 14, 2019: several corrections throughout, and new results at end of Chapter 11.