Algebraic topology is a subject central to mathematics and which,
at the level taught in a first course, can be considered as fully
mature. Thus, one expects, at least from a given viewpoint, a book
to be written that can be regarded as the definitive text and reference
of the subject. In my view, Allen Hatchers Algebraic topology
is such a book. Further-more, the book contains a wealth of additional
topics, examples and exercises that make it an inspiring book for
the motivated student.
...
To conclude, I strongly recommend Hatchers Algebraic Topology
to any serious student of mathematics with the only caveat being that
the book requires a certain level of mathematical sophistication.