Results 1 to 7 of 7

Math Help - about topological spaces

  1. #1
    Newbie
    Joined
    Mar 2009
    From
    São Paulo- Brazil
    Posts
    22

    about topological spaces

    Let Y be a subset of a topological space X. If any point of Y admits an open neighbourhood U in X such that U\cap{Y} is closed in U then Y is open in \bar{Y}.

    Thanks in advance.
    Last edited by Biscaim; June 28th 2009 at 07:23 AM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Hello,

    Okay, maybe I'm just rusty at topology, but what does it mean for a set Y to be open in another set, \bar{Y} ?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Mar 2009
    From
    São Paulo- Brazil
    Posts
    22
    Quote Originally Posted by Moo View Post
    Hello,

    Okay, maybe I'm just rusty at topology, but what does it mean for a set Y to be open in another set, \bar{Y} ?
    The closure of any subset of a topological space is a closed subspace.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Moo
    Moo is offline
    A Cute Angle Moo's Avatar
    Joined
    Mar 2008
    From
    P(I'm here)=1/3, P(I'm there)=t+1/3
    Posts
    5,618
    Thanks
    6
    Quote Originally Posted by Biscaim View Post
    The closure of any subset of a topological space is a closed subspace.
    I forgot to stress the word open.
    It's correct to say that a subset is closed/open in a topological space.
    But as far as I remember, I've never seen that a subset is closed/open in another subset...
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Super Member flyingsquirrel's Avatar
    Joined
    Apr 2008
    Posts
    802
    Quote Originally Posted by Moo View Post
    what does it mean for a set Y to be open in another set, \bar{Y} ?
    It means that Y is an open subspace of the topological space \overline{Y}, the topology on \overline{Y} being the topology induced by X on \overline{Y}. In other words, Y is open in \overline{Y} if and only if there exists a subset O of X which is open in X and such that Y=O\cap \overline{Y}.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Member
    Joined
    Jun 2009
    Posts
    113
    It is a nice problem. I'll use cl for closure and \cap for intersection
    and \subset for inclusion

    a) For each subset U of X open, U \cap cl(Y)\subset cl (U\cap Y).
    For proving this, take x in U \cap cl(Y) and let V be an open neighbourhood of x. U\cap V is an open neighbourhood of x, and since x is in the closure of Y, U\cap V \cap Y is nonempty. Thus V cuts U\cap Y therefore we conclude.

    b) Let y in Y. Take U an open neighbourhood of y such that U\cap Y is closed. From a) we conclude that U\cap cl(Y)\subset cl(U\cap Y)= U\ cap Y. Hence U\cap cl(Y)=U\ cap Y, and this means that each y in Y
    has an open neighbourhood in cl(Y) included in Y, i.e, Y is open in cl(Y)
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Member
    Joined
    Jun 2009
    Posts
    113

    Gap

    My argument in b) is not correct- CL(U\cap Y)=U\cap Y is not true

    We noly know that U\cap Y is only closed in U! Anyway we have

    a) U \cap cl(Y)\subset cl (U\cap Y)\cap U.

    Now b) is true because U\cap cl(Y)\subset cl(U\cap Y)\cap U= U\ cap Y
    Last edited by Enrique2; June 30th 2009 at 03:56 AM. Reason: giving the good argument
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Topological Spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: May 19th 2010, 08:14 PM
  2. Topological Spaces
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: March 4th 2010, 09:01 AM
  3. Hoeomorphic topological spaces
    Posted in the Differential Geometry Forum
    Replies: 1
    Last Post: March 19th 2009, 03:55 PM
  4. continuity in topological spaces
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: May 26th 2008, 01:12 AM
  5. question about topological spaces
    Posted in the Advanced Algebra Forum
    Replies: 0
    Last Post: May 19th 2008, 05:23 PM

Search Tags


/mathhelpforum @mathhelpforum