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Weakly sequentially compact supsets

  1. Feb 12, 2012 #1
    I need some help in understanding exactly the following definition (any links to sources will be great).

    What is boundedly weakly sequentially compact subset in the compact Hausdorf space with a regular Borel measure?

    What in this case will be the weak topology?

  2. jcsd
  3. Feb 13, 2012 #2


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    What is your background in math? I've never heard the terms "boundedly weakly sequentially" before. And I doubt the presence of a Borel measure has anything to do with it. So, to sum up my question,
    -Are you sure you've transcribed the question right? Because the "boundedly part in "boundedly weakly sequentially compact" sounds weird to me.
    -there is a lot of words in your question. Which ones do you understand and which ones not.
    -what is the context of the question/where are you coming from mathematically? Are you an engineer trying to read a math text, or an undergrad trying to understand a paper, etc?
  4. Feb 15, 2012 #3


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    Best I could come up with is:

    You have the weak topology (presumably in some function space), and a given set is
    sequentially compact in the weak topology. The weak topology on a collection of operators/functions, is the smallest topology that makes all the operators continuous; e.g., the product topology is the initial topology for the projection operators (note that the largest topology to make all continuous would be the uninteresting topology).

    A weak topology is also used when you have a collection of functions from a set --without a topology--into a topological space. Once can then (constructively) define a topology on the set so that this topology is the smallest topology (i.e. not contained in a larger topology) which makes the collection of all maps simultaneously continuous.

    There is a similar notion for final topology, in which you go in the reverse direction, mapping a space into a set, so that your final topology is the largest topology that makes all maps continuous.
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