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Math Help - simply connnected

  1. #1
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    simply connnected

    Hi,

    im trying to understand the meaning of " a region being simply connected" but i still cant get it.

    may i know what it means taht the complement of the z axis is not simply connected. the complement of the origin is simply connected....

    what does simply connected region means ( for stokes theorem to apply)

    thanks
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  2. #2
    Super Member Failure's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    Hi,

    im trying to understand the meaning of " a region being simply connected" but i still cant get it.

    may i know what it means taht the complement of the z axis is not simply connected. the complement of the origin is simply connected....

    what does simply connected region means ( for stokes theorem to apply)

    thanks
    What about looking it up Simply connected space - Wikipedia, the free encyclopedia

    note in particular the "informal discussion" on that page.
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  3. #3
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    Quote Originally Posted by Failure View Post
    What about looking it up Simply connected space - Wikipedia, the free encyclopedia

    note in particular the "informal discussion" on that page.
    i think i kind of get the idea of what a simply connected space is after reading the link..but can i know what is the complement of the z axis?

    is the complement of the origin the origin itself? hence any loop can always be contracted to the origin?

    i hope i have inferred it correctly..
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  4. #4
    Super Member Failure's Avatar
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    Quote Originally Posted by alexandrabel90 View Post
    i think i kind of get the idea of what a simply connected space is after reading the link..but can i know what is the complement of the z axis?

    is the complement of the origin the origin itself? hence any loop can always be contracted to the origin?

    i hope i have inferred it correctly..
    Note that it is important to mention the overall space that you are talking about. So, for example, if you remove the origin from \mathbb{R}^2 you get a space that is no longer simply connected. Whereas if you remove the origin from \mathbb{R}^3 you still have a simply connected space.

    Removing an entire line from \mathbb{R}^3 makes it no longer simply connected. For consider two points on the unit circle lying in the xy plane. The two arcs of the unit circle that connect them cannot be "deformed continuously" into each other, because that missing z axis gets in the way when we are trying to do so.
    On the other hand, if you remove only a ray from \mathbb{R}^2 or \mathbb{R}^3 the resulting space is still simply connected.
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