I would like to know some good problem books in various branches of undergraduate and graduate mathematics like group theory, galois theory, commutative algebra, real analysis, complex analysis, topology etc. The books should contain solution to many of the problems if not all, something similar to Problems in Algebraic Number Theory by M. Ram Murty, Jody Esmonde. Also, any text book contain solutions to many of exercises will also do.
Answer
The list of books that I would recommend to you,
Problems and solutions in mathematics, This book contains a number of questions and solutions on topics such as Group and Galois Theory.
General Topology:

V. Arhangel’skii, Fundamentals of General Topology .

K. Rao, Topology.
Differentiable Manifolds:

P. Gadea and J. Masqué, Analysis and Algebra on Differentiable Manifolds.

S. Morita, Geometry of Differential Forms.
Mathematical, Real and Complex Analysis:

D. Aliprantis and O. Burkinshaw,
Problems in Real Analysis. 
W. Kaczor, Problems in mathematical analysis.

R. Shakarchi, Problems and solutions for Complex analysis.

B.Makarov, Selected problems in real analysis.
and Schaum’s Outline of Theory and Problems are a series of supplementary texts as problems and solutions.
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