What is general topology good for?

[Arian.D]

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 Is point-set topology an active area of research in mathematics or it's outdated?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.

 

[lavinia]

As far as I know point set topology is not an active or at least important area of research.

In my opinion, it is a tool that is indispensable for analysis and geometry but by itself is not tremendously useful.

If you want to learn more topology try differential topology or complex analysis and Riemann surfaces.

There is also Algebraic Topology but I wouldn't tackle this next.

 

[theorem4.5.9]

You are learning the very basics of topology. I think that's kind of like learning calculus and then asking if calculus is an active area of research.

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.

I don't know much about physics or applied math, but as lavinia said, topology is in the background for several disciplines of mathematics.

Topology is a very active area of research, but you may have narrowly limited your definition of point-set topology to what you might find in Munkres. If so then you won't be satisfied with any answer (as you won't find any active research in virtually any textbook). Though I think low dimensional topology and knot theory should count (both very active) but I may be wrong.

As for topology being useful for physicists, I would venture to guess that every physicist should know some topology, even if informal, and only certain disciplines make great use of topology (like in gauge theories).

 

[mathwonk]

It is useful as words are useful. You cannot read any branch of mathematics after the most elementary without knowing basic topology. It is a formalization of the concept of continuity, one of the most useful and fundamental ideas in geometry, calssical physics, and mathematics.

 

[homeomorphic]

What are the important results that mathematicians have found using point-set topology?

It's used enough that you hardly think about things as "applications of point-set" topology after a while. It's just math.

Is point-set topology an active area of research in mathematics or it's outdated?

I wouldn't call it outdated, but it's not terribly active. Although, I knew a guy who did point set for his REU and the guy that he worked with did it, and there was some kind of conference, so evidently, there is something going on, still.

What are the other areas of mathematics that I could study after I've read point-set topology?

Mathwonk answered it.

How important point-set topology is for physicists?

It's important in proportion to how much math they use.

I'm not sure you could apply point-set directly to real life very much, but what happens is that you can apply it to say, differential equations, which can apply to real life. People are also attempting to apply more advanced topology in many different real life situations, which, again, build on point-set.

http://www.johndcook.com/blog/2010/09/13/applied-topology-and-dante-an-interview-with-robert-ghrist/

 

[Kreizhn]

At a very basic level, those topological properties that you listed are immensely important to almost all mathematics. Some simple but super useful examples are the extreme value theorem (essentially a corollary to the fact that the continuous image of compact sets are compact) and the Banach-Alaoglu theorem (which uses Tychonoff's theorem). A ridiculous number of results in differential topology, algebraic topology, algebraic geometry, etc come from these basic results also, and the applications of those fields are immense.

Is point-set topology an active area of research in mathematics or it's outdated?

I have a good friend whose research in is point-set topology, though I believe his father was also such a point-set topologist so maybe it was passed down genetically? They are interesting characters, those point-set topologists, and very brilliant. They have a very unique way of thinking and do mathematics that even the rest of us seem to think is bizarre. And while we in my (very limited) experience, many people researching pure mathematics tend to have pragmatic crises, I do find that the point-set topologists experience this to a greater degree .

 

[homeomorphic]

In a sense, there is a way in which point set is very applicable. You can use it to prove results in calculus. However, it is not necessary to use it.

I like to think of the operator norm of a linear map of finite dimensional vector spaces in terms of compactness (since the unit ball is compact, the map attains a maximum on it, which is exactly the operator norm). That's kind of useful, and I think it sidesteps a bit of calculation.

Another thing about point-set is that I think it helps your higher-dimensional intuition/comfort level. The reason why is that, in point-set, you think about a space as a set of points that might not even have a well-defined dimension and you think about the topology of those spaces and get an intuition for them. If you get intuition about those, suddenly, an n-dimensional manifold is a little bit less scary, since it's a special case of that. So, it kind of builds your ability to think abstractly.

 

[Kreizhn]

Another thing about point-set is that I think it helps your higher-dimensional intuition/comfort level.

This is definitely true. Alternatively, point-set can also destroy your intuition, which is always fun since you glean lots of cool math when these things happen. I think my favourite example of this is the Zariski topology on the spectrum of a ring. Points are not closed in this space, and if we introduce coordinates, an irreducible curve is a subset of every point contained within that curve.