Math Is Fun Forum

bossk171

Member

Registered: 2007-07-16

Posts: 305

Topology Headache

I'm learning topology from Elements of Point Set Topology by John D. Baum. I'm pretty sure at this point that I hate it.

Some questions:

What's the difference between Point Set Topology and Algabraic Topology?

Can someone kindly explain to me what indices are? The book show the notation, but doesn't really explain it. Some examples would be great.

Also, sometimes it says something like "prove that A is a subset of B does not imply that B is a subset of A" How would I prove that, isn't it just plain obvious? Am I supposed to be providing a counter example?

And super bonus points to anyone who can explain to me why I shouldn't just give up now and do something easier.

I miss calculus.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Topology Headache

What's the difference between Point Set Topology and Algabraic Topology?

Point-set topology can be viewed as a generalization of real-analysis, where you study a space of points and sequences/functions on them.  Algebraic-topology is using the tools of algebra to study (point-set) topological spaces.

Can someone kindly explain to me what indices are? The book show the notation, but doesn't really explain it. Some examples would be great.

Indices is the plural of index, so I'm not entirely sure what it is you're referring to.  My best guess is that a lot of times in topology you use an indexing set.  For example, if I have an open set O (in the reals), and a point alpha is in O, then there exists an open ball around alpha contained in O, denoted by U_\alpha.  Then

Now my collection of these balls is indexed by alpha.  Is this what you mean?

Am I supposed to be providing a counter example?

Yes.

And super bonus points to anyone who can explain to me why I shouldn't just give up now and do something easier.

I miss calculus.

Because all math is hard and it will fight you.  If you have patience, you'll get it.  The great thing about mathematics is that the only way you can ever really lose is to throw in the towel.

Why are you studying topology in the first place?  Do you plan to go on to grad school?  Or are you simply interested in the subject?

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Topology Headache

Indices is the plural of index, so I'm not entirely sure what it is you're referring to.  My best guess is that a lot of times in topology you use an indexing set.  For example, if I have an open set O (in the reals), and a point alpha is in O, then there exists an open ball around alpha contained in O, denoted by U_\alpha.  Then

Now my collection of these balls is indexed by alpha.  Is this what you mean?

Yes, that's what I mean. I'm still not really sure what they mean though. Is there an example you could give, I think it's all the theory without any examples that's overwhelming me.

Why are you studying topology in the first place?  Do you plan to go on to grad school?  Or are you simply interested in the subject?

There's a Prof. at my school that offered to teach me complex analysis (no credits or anything, just one on one for fun). I go to a community college and the highest level math they teach is Calc 2 (I took the equivalent in high school). Being a fan of calculus and complex numbers, I thought that'd be the coolest thing ever, but just a few weeks in I'm already fried out. But I agree about sticking with it, yesterday's post was a moment of weakness, hopefully this will start to clear up as I move along.

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Topology Headache

Being a fan of calculus and complex numbers, I thought that'd be the coolest thing ever, but just a few weeks in I'm already fried out.

Keep going.  I am not a fan of calculus (for the most part), and I found it was really cool.  Your professor is entirely right of course, you do need a good background in topology when learning complex analysis.

Not having a background in proofs however, things may get a bit hard.  One thing to keep in mind is that it is supposed to be frustrating... One of my professors described point-set topology as "...the gates of Hell.  Everything you thought was true turns out to have a counter-example."

Is there an example you could give, I think it's all the theory without any examples that's overwhelming me.

Unfortunately that is an example... and this is something you're going to have to get used to over time.  In more advanced mathematics, examples can (at times) be simpler theorems...

But let's start slower.  Let's say I have a set {1, 2, 3}.  When I index one set with another set, what I do is a set up a 1-1 and onto correspondence.  So for example, I can index {1, 2, 3} with the set {x_a, x_b, x_c}.  So now:

x_a means 1
x_b means 2
x_c means 3

And in this example, what I've just done is a bit silly.  There is really no need to index this set, or at least I can't think of one.  But when you go to bigger sets, indexing makes referring to elements a lot easier.

Let's go to the real line.  Now I'm doing a problem, and for every integer I want to associate with it an interval of size 1/2.   So for example, I associate the interval (-1/2, 1/2) with the integer 0, and the interval (1/2, 3/2) with the integer 1.  Now the number 0 corresponds to my open set (-1/2, 1/2).  But it would be just awkward and confusing if I were to just write the number 0 for my interval.  So I introduce a letter, say U, by which I will index with my number 0.

So for example:

U_0 = (-1/2, 1/2)
U_1 = (1/2, 3/2)
....

Now writing U_15 is much easier than writing (29/2, 31/2), and much more clear to the reader.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

bossk171

Member

Registered: 2007-07-16

Posts: 305

Re: Topology Headache

That little bit cleared up almost everything for me, Thanks! Now I'm thinking of indices like I think dictionaries in Python, is that correct?

Just one last question, how would you relate your example:

to this notation:

Last edited by bossk171 (2009-05-31 03:49:04)

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There are 10 types of people in the world, those who understand binary, those who don't, and those who can use induction.

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Topology Headache

And just to wet your appetite, here is something you'll probably see in complex analysis, called a residue integral.

You use complex analysis to calculate the value of that integral.  Another cool theorem is known as Liouville's theorem:

If f is a complex differentiable function defined on the entire complex plane and f is bounded, then f is a constant function.

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."

Ricky

Moderator

Registered: 2005-12-04

Posts: 3,791

Re: Topology Headache

That little bit cleared up almost everything for me, Thanks! Now I'm thinking of indices like I think dictionaries in Python, is that correct?

Correct.  Just remember that the name you call something is important.  U_0 conveys information about what thing I'm talking about, here the interval located around the point 0.

Just one last question, how would you relate your example:

to this notation:

I would write:

An alternate way to write this set is:

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"In the real world, this would be a problem.  But in mathematics, we can just define a place where this problem doesn't exist.  So we'll go ahead and do that now..."