Results 1 to 4 of 4

Math Help - Open Ball

  1. #1
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782

    Open Ball

    Prove that a subset of a metric space is open iff it is a union of open balls.
    So I did this:

    Suppose a subset of a metric space is open.
    Let (X,d) be a metric space.
    Let U \subset X
    Then there exists some \varepsilon >0 such that B_{\varepsilon}(x;d) \subset U.

    But x is an arbitrary element of U.
    Therefore centre a ball around every point in U.
    \cup_{x \in U} B_{\varepsilon}(x;d)=U.

    (Since the centre of each of these balls is a point in U, the union of them must be the entire set U.)

    Conversely, suppose a subset U of a metric space X is \cup_{x \in U} B_{\varepsilon}(x;d)=U.

    Then by definition U is open.

    Is this right?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,963
    Thanks
    1784
    Awards
    1
    Actually it is not correct. The idea is right but not the details.
    If Uis an open set then \left( {\forall x \in U} \right)\left( {\exists \varepsilon _x  > 0} \right)\left[ {B\left( {x;\varepsilon _x } \right) \subseteq U} \right].
    Now you can say that U = \bigcup\limits_{x \in U} {B\left( {x;\varepsilon _x } \right)} .

    The other way is trivial.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member Showcase_22's Avatar
    Joined
    Sep 2006
    From
    The raggedy edge.
    Posts
    782
    Is there any significance to the fact that, in the definition you've provided, you've put the part about the ball in square brackets and not "()"?

    I know it's a bit trivial, but it looks like a useful thing to know.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor

    Joined
    Aug 2006
    Posts
    18,963
    Thanks
    1784
    Awards
    1
    Quote Originally Posted by Showcase_22 View Post
    Is there any significance to the fact that, in the definition you've provided, you've put the part about the ball in square brackets and not "()"? I know it's a bit trivial, but it looks like a useful thing to know.
    I don't understand your question.
    There is no standard to write the notation for balls.
    In logical notation I use () for the quantifies and [] for the resulting proposition.
    The point I made is that you used a fixed \varepsilon .
    That is incorrect. Each x\in U corresponds to its own <br />
\varepsilon_x
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. open ball
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 19th 2011, 12:34 PM
  2. open ball metric space
    Posted in the Differential Geometry Forum
    Replies: 0
    Last Post: November 1st 2010, 12:39 PM
  3. linear map open iff image of unit ball contains ball around 0
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: May 25th 2010, 02:15 AM
  4. Closure of open ball
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: March 28th 2010, 02:39 PM
  5. Diameter of an Open Ball
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: December 14th 2009, 05:34 PM

Search Tags


/mathhelpforum @mathhelpforum