I will just start reading topology but I am confused regarding:
- Point topology-Stephen Gaal
- Algebraic topology- Hatcher
- Combinatorial and Differential topology-Prasalov
- 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.
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!
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! ^_^