I know that my question is not very clear, so I'll try my best to clarify it. Firstly, by general topology I mean point-set topology, because that's the only form of topology that I've encountered so far. In point-set topology, they teach us a lot of new definitions like open sets (that are defined as members of a topology on a set X, assuming that we define a topology by open sets axioms), closed sets, limit points, interior points, exterior points, boundary points, etc, then we learn how to create new topological spaces like the product space and how to induce a topology on subsets of X to form subspaces of a topological space and then we learn about continuous maps and properties of a space that are invariant under invertible continuous maps (homeomorphisms) that we define such properties as topological properties. Well, I get the idea, topology is the study of topological properties of space. Intuitively speaking, since distance is not a topological property, topology could be interpreted as a branch of mathematics that studies the shape of mathematical objects without caring about their size and other distance-related properties. I've learned many theorems in topology, but still I don't know what these theorems are good for and where I could apply them in real life. Yes, I could use these theorems to solve other problems in general topology, but I can't predict how topology, I mean point-set topology, could interact with other branches of mathematics. My question could be precisely said in this way: What are the important results that mathematicians have found using point-set topology? Is point-set topology an active area of research in mathematics or it's outdated? What are the other areas of mathematics that I could study after I've read point-set topology? How important point-set topology is for physicists? These are my questions. I hope my question isn't vague.