Overcoming Abstraction in Mathematics

[Avatrin]

Hi

As I am venturing in graduate level mathematics, I am having a recurring problem; I keep getting stuck in the abstraction of it. Usually it involved set theory; I never get "fluent" in it. However, the main problem is abstraction.

For instance, this semester I had topology, and the curriculum was from chapters 1, 2, 3, 4, 7 and 9 in Munkres. I was stuck in chapter 2 for ages. The fundamental group was no big problem since it is very visual. Metric spaces and function spaces were not much of a problem either. However, the biggest problem I had was the quotient space. I never got through the section about it. I could not get further than a few pages, although I tried.

I seem to be able to think in terms of groups, metric spaces and even specific topological spaces. However, when chapters get more abstract than that, the book loses me completely.

All of my knowledge about set theory comes from introductory chapters in books about metric spaces, algebra and topology. Should I read a book on set theory specifically?

Should I just move onto more intuitive subjects like partial differential equations? Or, is there a way to overcome this problem?

 

[FactChecker]

If you have no problem with groups, topological spaces, and function spaces, then you can handle abstraction. A lot of abstract things seem confusing on the first encounter but become absolutely clear to you later. Quotient spaces will be like that. They are like a projection of a higher dimensional space onto a subspace, where any two elements that project to the same point are considered equivalent. The subspace is the quotient space.

That being said -- if you think you have a weakness in set theory, you should fix that. You don't have to go deeper into it than you need to become comfortable.

 

[Avatrin]

Yeah, but, groups and function spaces are generalizations of things I already know. However, something like quotient spaces do not seem to have that. I have encountered quotient groups in abstract algebra, but they are very different.

 

[fresh_42]

However, the biggest problem I had was the quotient space. I never got through the section about it. I could not get further than a few pages, although I tried.

Can you tell something specific? I mean the remainders of integer divisions are quotient spaces, "three third of a cake is a whole cake" is an equation in a quotient space, the description of the image of a projection is a quotient space, not distinguishing between equivalent objects means handling elements of a quotient space, e.g. the hours on the clock on the wall.

 

[lavinia]

Yeah, but, groups and function spaces are generalizations of things I already know. However, something like quotient spaces do not seem to have that. I have encountered quotient groups in abstract algebra, but they are very different.

It is always helpful to work through examples. I can give you some for quotient spaces. Generally abstractions derive from many examples that have common properties. So the examples give you intuition.

 

[Avatrin]

Can you tell something specific? I mean the remainders of integer divisions are quotient spaces, "three third of a cake is a whole cake" is an equation in a quotient space, the description of the image of a projection is a quotient space, not distinguishing between equivalent objects means handling elements of a quotient space, e.g. the hours on the clock on the wall.

I think you are talking about quotient groups from algebra. Those I had no problems with. I am talking about a quotient space in topology induced by the quotient map.

Of course, that is unless they are the same thing. The issue with that is that Munkres states that the quotient map maps saturated open sets to saturated open sets, and that sounds more general than treating equivalence classes as elements as is the case in algebra.

Also, this happens a lot. Quotient spaces is just the one that annoyed me the most this semester. I overcame the abstraction in algebra by reading Stillwell's Elements of Algebra (by focusing on polynomials it tied modern algebra to something I am familiar with). I am looking for something similar for topology.

 

[HallsofIvy]

Essentially, If you have topological space A with subspace B, the "quotient space", A/B, is the result of "contracting" subspace B to a single point. If B is a connected space, say, $A= R^2$ and B is the unit disk, that is very simple. If B is not so simple, say, B is the unit circle it gets more complicated.

 

[FactChecker]

I think you are talking about quotient groups from algebra. Those I had no problems with. I am talking about a quotient space in topology induced by the quotient map.

So the real problem is in understanding the topological properties of the quotient space. There may be simple examples that are familiar and should give you some intuition. (Like "gluing" the circumference of a disk to a single point to form a spherical surface.) I suggest you look at those till you are more comfortable with the concepts.

PS. It may also be that these concepts are just more difficult than you expected them to be. You are probably not the only person who has trouble with them.

 

[Avatrin]

So the real problem is in understanding the topological properties of the quotient space. There may be simple examples that are familiar and should give you some intuition. (Like "gluing" the circumference of a disk to a single point to form a spherical surface.) I suggest you look at those till you are more comfortable with the concepts.

PS. It may also be that these concepts are just more difficult than you expected them to be. You are probably not the only person who has trouble with them.

I did something I reckon is similar; Gluing the circumference of two n-dimensional discs, and deforming them into a n+1-dimensional sphere is something I had to do for homework. I did it correctly, but I never used the language of quotient spaces (I hadn't even read the section on quotient spaces). The idea was simply too intuitive for me to bother reading the relevant chapters (I, after all, knew the concept of equivalence classes from algebra).

Also, I just decided to read the definition of a quotient space without having read the page before, and I have to rephrase my problem. That idea seems more intuitive than what comes before in Munkres; What I am truly struggling with is the quotient topology, and the quotient map. In fact, the quotient topology is not that bad either, except it depends on the quotient map. That's where the meat of the problem lies.