Topology: Nets and filters

  1. 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,

  2. jcsd
  3. marcus

    marcus 24,218
    Science Advisor
    Gold Member
    2014 Award

    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


    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
  4. 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
    Last edited: Jul 17, 2003
  5. marcus

    marcus 24,218
    Science Advisor
    Gold Member
    2014 Award

    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)
    Last edited: Jul 17, 2003
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