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Math Help - What is a neighborhood?

  1. #1
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    What is a neighborhood?

    Is it:

    a) a set A that has a subset B with a point P in it (in which case, the point P's neighborhood is set A)

    b) "Set V is a neighborhood of point P if a small disk around P is contained in V" - Wikipedia. What is this small disk? Is it a set? Is there an established radius?
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  2. #2
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    Quote Originally Posted by lrl4565 View Post
    Is it:

    a) a set A that has a subset B with a point P in it (in which case, the point P's neighborhood is set A)
    No, B has to be open. And in which case, A is one of P's neighbourhoods.

    b) "Set V is a neighborhood of point P if a small disk around P is contained in V" - Wikipedia. What is this small disk? Is it a set? Is there an established radius?
    No established radius. This definition only holds for metric spaces.
    It's more commonly called a ball
    By the way, in Wikipedia, they talk about a ball, not a disk...
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  3. #3
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    [QUOTE=lrl4565;502367]Is it:

    a) a set A that has a subset B with a point P in it (in which case, the point P's neighborhood is set A)


    Quote Originally Posted by Moo View Post
    No, B has to be open. And in which case, A is one of P's neighbourhoods.

    I agree that B needs to be open, but there is some dispute between the exact definition of a neighborhood. Some French mathematicians following Bourbaki sometimes define a neighborhood of a point to be any set containing an open set containing that point, whereas more often a neighborhood is just an open set containing the point.
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    Quote Originally Posted by Drexel28 View Post

    I agree that B needs to be open, but there is some dispute between the exact definition of a neighborhood. Some French mathematicians following Bourbaki sometimes define a neighborhood of a point to be any set containing an open set containing that point, whereas more often a neighborhood is just an open set containing the point.
    That's what your Wikipedia says... I also read the Spanish Wikipedia, which states the same. So I'd like to know where your red "more often" comes from !!

    Maybe you're confusing with the fact that an open set is a neighbourhood of all its points.
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    Quote Originally Posted by Drexel28 View Post

    I agree that B needs to be open, but there is some dispute between the exact definition of a neighborhood. Some French mathematicians following Bourbaki sometimes define a neighborhood of a point to be any set containing an open set containing that point, whereas more often a neighborhood is just an open set containing the point.
    That's what your Wikipedia says... I also read the Spanish, German & Italian Wikipedia, and they write the same definition. So I'd like to know where your red "more often" comes from !!

    Maybe you're confusing with the fact that an open set is a neighbourhood for all of its points.
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    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Moo View Post
    That's what your Wikipedia says... I also read the Spanish Wikipedia, which states the same. So I'd like to know where your red "more often" comes from !!
    I'm not saying it's the world standard, I'm saying that in most of the 32 topology books I own that is how a neighborhood is defined.

    Maybe you're confusing with the fact that an open set is a neighbourhood of all its points.
    I hate this expression, it personally aways confuses me. This is the approach taken by Bert Mendelson in his "Introduction to Topology" and otherwise good book.
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  7. #7
    Moo
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    Defining it this way is more laziness than something else
    You're just skipping the 'useless' part of the definition !

    So tell me, out of 32 of these books, how many define it in the second way ?
    And out of these 32 books, how many are written by American authors ?
    American school is not alone in the world :P

    I'm sorry I'm afraid I can't decrypt Russian, so I don't know what the 'Russian standard' is, according to Wikipedia.

    I hate this expression, it personally aways confuses me.
    It's a property that is always true. Even if it confuses you, it won't prevent people from using it.
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  8. #8
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by Moo View Post

    So tell me, out of 32 of these books, how many define it in the second way ?
    3:

    General Topology- Bourbaki Parts I and II

    Introduction to Topology-Bert Mendelson

    And my favorite professor

    Topological Methods in Euclidean Space-Greg Naber

    And out of these 32 books, how many are written by American authors ?
    Most

    American school is not alone in the world :P
    Thus me saying it's not a world standard. Plus, the OP is from America (supposedly)
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  9. #9
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    Thus me saying it's not a world standard. Plus, the OP is from America (supposedly)
    And so what ? I'm correcting the definition he wrote. And the way he wrote it, it is more likely the first than the second one !!!
    Slow it down, create a new thread, it's not the topic here !
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    Alright, what I've gathered:

    Hypothetical situation:

    Set A: [1,10]
    Set B: (3, 7)
    Point P: 6

    Both Set A and Set B are neighborhoods of point P
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  11. #11
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by lrl4565 View Post
    Alright, what I've gathered:

    Hypothetical situation:

    Set A: [1,10]
    Set B: (3, 7)
    Point P: 6

    Both Set A and Set B are neighborhoods of point P
    That depends on your definition once again and what you call "open".

    If I were a teacher and I was teaching you about neighborhoods with the usual real number structure than I would say B is a neighborhood but A isn't.
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  12. #12
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    So, there's dispute over the exact definition of a neighborhood. I am American. What is the most common?

    What IS a neighborhood? Why categorize subset B (with point P inside) of set A as the neighborhood of P? Labeling purposes, or does it mean something more?
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  13. #13
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    Generally, anything you would want to do with neighborhoods you could also do with open sets. You can develop topology by, instead of declaring certain sets open, defining a filter of neighborhoods for every point and everything comes out the same. It is my impression that open sets are used because they are somewhat simpler to work with, though I've never really used the definition through neighborhoods for anything, so I can't say.
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