Point-set & Low Dimensional Topology

Below is a list of reviews of books on point-set topology.

Notes on Introductory Point-Set Topology
Hatcher, A. (2005). Free Download
Prerequisites: Basic set theory.

Covers essential material in point-set topology over the course of about 40 pages, geared towards introducing the constructions and material needed for Hatcher’s algebraic topology textbook. The notes are fairly chatty and more focused on giving the reader the bare minimum they need from point-set topology for algebraic topology, commutative algebra, or other fields.

Topology: A Categorical Approach
Bradley, T-D., Bryson, T., Terilla, J. (2020). ISBN 9780262539357.
Prerequisites: comfort with sets and basic proof-based mathematics.

This book is quite different from the others on this list, and might best be used in conjunction with one of the others. The book develops basic category theory and point-set topology at the same time, with a different and more modern perspective than point-set is usually introduced. Potentially obscure topological definitions are shown to be very natural and straightforward through their universal properties.

Lectures on the Topology of 3-Manifolds
Saveliev, N. (2nd ed. 2011) ISBN 9783110162721.
Prerequisites: manifolds, standard algebraic topology (homology, coverings…), and a bit of knot theory

It has a very clear introductive glossary that recalls all the notions used in the book, as well as notations. It studies 3-manifolds through Kirby calculus, as well as the boundaries of 4-manifolds, to then dive into more technical invariants (the goal being to study the Casson invariant). The second edition has much needed additions.

4-manifolds and Kirby calculus
Stipsicz, A., Gompf, R. (1999) ISBN 9780821809945.
As for background, it is preferrable to already be familiar with most of the stuff discussed in the book, on top of standard topology material

This book is one of the best, if not the best, in its field. However, it cruelly lacks a thorough exposition of the material, despite being already ~600 pages long. This covers lots of topics, and has the amazing benefit of giving a lot of examples, and in different points of view. It has a lot of drawings, at the cost of sketching a lot of the proofs.