I will just start reading topology but I am confused regarding:

  1. Point topology-Stephen Gaal
  2. Algebraic topology- Hatcher
  3. Combinatorial and Differential topology-Prasalov
  4. Topology - Munkres

I am quite confused are they different topics? Note:I searched but but I cannot understand the difference. And which topic should I start first.

  • $\begingroup$ For 1. you could go to General Topology by Engelking, or parts of it. It's big and has much extra material within the Problems & Exercises. I had Introduction To Topology And Modern Analysis by Simmons on a course, & I like it. You should have some basic Set Theory, e.g. on cardinal arithmetic. The small introductory Set Theory by Suppes is good. $\endgroup$ – DanielWainfleet Oct 21 at 17:28

Broadly, topology studies notions of "closeness" without having a notion of "distance". We formalize this using things called "open sets". It turns out this definition is extremely flexible, and admits many objects which are not obviously geometric. In fact, topological spaces can have extremely counterintuitive properties in general!

  • Point Set Topology, then, is the study of all topological spaces. This subfield often has an emphasis on various extra assumptions we put on our topological spaces in order to control their behavior. This subject is often combinatorial (and has close ties to set theory), and is the bedrock of the rest of "the topologies". It is extremely important to have a working knowledge of some of the information from this area in order to do anything else topological. Notions such as Hausdorff, Compact, Second Countable, Connected, etc. are very common assumptions which we make when studying geometric objects. Understanding this field helps you understand what can go wrong when trying to prove theorems in the other areas. Of course, there are many subtleties which will be totally irrelevant in the other areas - most people are happy to assume their topological spaces are "nice enough" for their area, and will never need to know what happens with spaces "less nice" than the ones they're used to.

Once we have the set-theoretic issues out of the way, we can start doing geometry. If you've heard topology described as "rubber-sheet geometry", this is what you're being sold on. I remember I was upset when I first started learning about point set topology, because it seemed so removed from what I was promised. Thankfully, as I mentioned earlier, you don't really need much point set topology in order to start using it! Broadly, this branch of topology falls into two categories:

  • Differential Topology studies topological spaces that are "nice enough" to do calculus. When studying these objects, calculus is a fundamental tool, and some of the theorems can feel "analytic" in nature. Of course, algebra is always just under the surface, and it seems the subject is only becoming more algebraic with time. A typical assumption in this area might be "every point has an open neighborhood that looks like $\mathbb{R}^n$". You can see how this one assumption prevents a lot of the fine structure of point-set topology from being relevant! However, working in this area requires a knowledge of linear algebra and calculus to get started.

  • Algebraic Topology studies invariants of topological spaces. Algebraic Topologists tend to work with a broader class of spaces than the Differential Topologists, and they don't have the power of calculus as a result. Instead, algebraic topologists try to understand the structure of the space by looking at (say) all of the ways a sphere can map into it. A typical assumption in this are might be "path connected, locally path connected, and semilocally simply connected". While these conditions are more technical than before, they still prohibit a lot of the behavior that point set topologists might study. This subject also has a combinatorial flavor, and is the source of a lot of powerful nonexistence results. In exchange for the calculus-flavored tools that we're leaving behind, an algebraic topologist should be familiar with groups and rings, as well as category theory.

Obviously this is only a vague birds-eye view of the topics, and there is constant interplay between algebraic and differential topology (as well as other fields). To answer your implicit question, a very standard approach is to first learn some point set topology (though perhaps not worrying too much about the details of, say $T_0$ spaces that are not $T_1$), and then move on to either differential or algebraic (or both!). Munkres and Dugundji are the standard point set books, but they can both be expensive. I've heard good things about Jänich, though I admit I haven't read it myself.

Importantly, I would argue that you should work through a more geometric topology book at the same time as your point set one. At the risk of offending any point set topologists in the room, most people find it rather dry, and it's nice to keep sight of where you're going. As a perhaps offbeat recommendation, I'm very fond of Henle's "A Combinatorial Introduction to Topology". It is extremely approachable, and does a fantastic job explaining basic differential and algebraic topology by means of combinatorial arguments. It is short and lucid, and I'm extremely lucky to have found it at a used bookstore a while ago, because I don't think I would have read it otherwise. It also exists as a dover paperback, making it pleasantly affordable!

Once you're ready for something heavier, I like Tu's "Introduction to Manifolds", as well as Hatcher's "Algebraic Topology". Unfortunately, I read both of these after I had already been exposed to a lot of topological ideas elsewhere. Because of this, I can't speak to how they work as a book for complete beginners. I suspect any good recommendation of a more advanced book will require a more thorough understanding of your current background, but maybe some commenters or other answers will provide insight into some introductory books in these areas.

And, of course, caveat lector: Topology is a deep and broad branch of modern mathematics with connections everywhere. What I've explained in this answer is only the tip of the iceberg, and I'm sure there are many mathematicians would choose different "main ideas" and different "example hypotheses" in the above descriptions. Hopefully someday soon you will have learned enough to have opinions of your own!

I hope this helps! ^_^

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  • $\begingroup$ I love point-set topology and it $is$ rather dry, but so am I. $\endgroup$ – DanielWainfleet Oct 21 at 17:33
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    $\begingroup$ Nothing wrong with dry! I'm a logician ;) $\endgroup$ – HallaSurvivor Oct 21 at 17:36

All four books you mention work with topological spaces.

If you begin studying topology, you will probably want to start with what is called point-set topology. This just means the part of topology that is concerned with some fundamental notions such as compactness, connectedness, ... You study properties of topological spaces (all formulated in terms of points and sets), without associating fancy objects to them.

Algebraic topology associates fancy objects to topological spaces (groups, rings) and studies the spaces through those algebraic (hence the name) objects.

Combinatorial topology is an old name for algebraic topology: some of those algebraic objects (the homology groups) have combinatorial descriptions/constructions.

Differential topology is a combination of algebraic topology and differential geometry. It is for example concerned with the relation between homology groups (purely topological notion) and the de Rham cohomology (differentiable notion).

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