What are the differences between the filter and net approaches in topology?

In summary, Dimitri Terryn is planning to double major in physics and math and has started studying topology over the summer. He has been informed that there are two approaches to the subject, one involving filters and the other involving nets. His topology professor for next year favors the filters approach, but Dimitri's friend has told him that they use the net approach. Dimitri asks for an explanation of the difference between the two approaches and recommendations for online textbooks on the filters approach. The conversation also touches on the definitions of filters, convergence, compactness, and open and closed sets.
  • #1
Kalimaa23
279
0
Greetings,

Having decided on a physics/math double major next year, I decided to get a head start this summer.

After tackling some classical mechanics, my next target is topology.

My problem is the following: I have been informed that there are two approaches to the subject, one involving filters and the other one involving nets (note that these terms are directly translated from Dutch, if other terms are used in English I apologize).

I've been told my topology professor for next year favours the filters. I have found some textbooks online, but a friend has told me they use the net approach.

My question is this: could somebody explain to me the difference between the two approaches, without going into too much detail. Also, it would be greatly appreciated if someone could point me in the direction of any online textbooks that deal with the filters approach.

Thanks in advance,

-Dimi
 
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  • #2
Originally posted by Dimitri Terryn

After tackling some classical mechanics, my next target is topology...

My problem is the following: I have been informed that there are two approaches to the subject, one involving filters and the other one involving nets...

I've been told my topology professor for next year favours the filters...

Filters and nets are only a part of the story, not the whole of topology. My idea is to get whatever book you can and start with it. Dont worry so much about whether your first book takes exactly the same approach as your professor.


This might help get you started. A FILTER is just a generalization of the idea of convergence to a limit. You know about convergent sequences of real numbers and you probably can define the LIMIT of a sequence.

Here are some definitions:

-------------------------
In fact I believe these are the three main definitions about filters!

A FILTER is a family F of non-empty sets which satisfy the condition: If A,B are sets in F, then there is a set C in F which is contained in A intersect B.

A filter F is said to CONVERGE to the point x provided each neighborhood of x contains a member of F.

A space is said to be COMPACT provided each filter consisting of closed sets is contained in a convergent filter.

-------------------------
Here are an auxilliary definition which the Prof may or may not use, and another way of defining compactness:

A filter F is said to CLUSTER at the point x provided each neighborhood of x intersects a member of F.

A space is compact if and only if each filter consisting of closed sets clusters at some point.
-------------------------

Now let us be realistic. The definitions probably sound very abstract when one hears them for the first time! So think of a sequence of numbers

{an}

And construct a family of sets F = {AN} which are just
the "tails" of the sequence
AN = {an for all n > N}

F is now a set of sets of numbers
F is the set of "tails" of the original sequence

Check that F is a filter.

The idea that the original sequence converges to a number x is the same as the idea that the filter F converges to x.

--------------------

What is clever about the idea of a filter?
The generality. A filter does not have to be countable
and it does not have to have an index set like 1,2,3,4,...
The set of sets take care of ordering themselves by inclusion.
It can be used for other things. It is in some way the most
simple and general picture of convergence, with the least
unnecessary detail.

However, because it looks weird and strange when you first see it, no one can at first understand why the professor should introduce such an idea and there will be 2 or 3 days of confusion.

-----------------------

The first definitions of topology are the "open sets"
The space X has a family of open sets satisfying the condition
that the union of open sets is again open
and the finite intersection also.

And the "closed" sets are the complements of open sets.

The intuitive idea of a closed set is one which includes its border.
But all this is being defined without the help of a metric or distance! Thus one has these abstract definitions in order to finesse things like "border" and "converges to a point" and "continuous function" which would be very easy to describe if one had the help of a distance measurement.

Go ahead and learn about open and closed sets. Any online textbook for pointset or general topology should be useful.
Nets and filters are no big deal and should not make you wait
at the door
 
  • #3
Thanks for the info. From your examples I can see now that the professor in question (she thought our freshman multivariable calculus course) has already nudged us towards the filter idea, if subtly...

Mathematicians, devious they are:wink:

Edited to account for blatant spelling errors
 
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  • #4
Originally posted by Dimitri Terryn
Thanks for the info. From your examples I can see now that the professor in question (she taught our freshman multivariable calculus course) has already nudged us towards the filter idea, if subtly...

Mathematicians, devious they are:wink:

Aesthetics is basic in mathematics and this professor sounds like she might have good taste. If so, my compliments on your good fortune.

(edited to correct for tendency to wander off topic)
 
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1. What is the difference between a net and a filter in topology?

A net and a filter are both mathematical tools used in topology to study the convergence of sequences in a topological space. The main difference between them is that a net is a function from a directed set to a topological space, while a filter is a subset of the power set of a topological space. In simpler terms, a net assigns points in a topological space to a particular index, while a filter is a collection of subsets that satisfy certain properties.

2. How do nets and filters relate to sequences in topology?

Nets and filters are generalizations of sequences in topology. A sequence can be seen as a special case of a net, where the index set is the set of natural numbers. Similarly, a filter can be seen as a collection of sets that contain infinitely many elements of a given sequence. Nets and filters provide a more flexible and powerful framework for studying the convergence of sequences in topological spaces.

3. What is the significance of directed sets in nets and filters?

Directed sets play a crucial role in defining nets and filters. A directed set is a partially ordered set in which every finite subset has an upper bound. This allows for a more general definition of convergence in topological spaces, as it does not rely on the traditional notion of a sequence having a limit point. Directed sets allow for a more abstract and flexible approach to studying convergence in topological spaces.

4. How are nets and filters used in topological proofs?

Nets and filters are powerful tools in topological proofs, particularly when dealing with convergence and continuity. They allow for a more general and abstract approach to studying properties of topological spaces. In particular, they are useful in proving theorems related to compactness, connectedness, and continuity.

5. Are there any real-world applications of nets and filters?

While nets and filters are primarily used in the field of topology, they have also found applications in other areas of mathematics, such as functional analysis and probability theory. In real-world applications, they can be used to study the behavior of systems that involve a large number of variables, such as in economics or physics. They are also used in computer science, particularly in the design and analysis of algorithms and data structures.

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